{"id":2840,"date":"2024-02-09T19:31:55","date_gmt":"2024-02-09T19:31:55","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/?post_type=chapter&p=2840"},"modified":"2024-03-08T23:57:26","modified_gmt":"2024-03-08T23:57:26","slug":"5-4-introduction-to-equations","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/chapter\/5-4-introduction-to-equations\/","title":{"raw":"5.4 Introduction to Equations","rendered":"5.4 Introduction to Equations"},"content":{"raw":"
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Learning Objectives<\/h3>\r\n
    \r\n \t
  • Explain what an equation in one variable represents.<\/li>\r\n \t
  • Determine if a given value for a variable is a solution of an equation.<\/li>\r\n \t
  • Classify an equation as conditional, a contradiction or an identity.<\/li>\r\n<\/ul>\r\n<\/div>\r\n
    \r\n

    Key words<\/h3>\r\n
      \r\n \t
    • solution<\/strong>: a value that can be substituted for a variable to make an equation true.<\/li>\r\n \t
    • unknown<\/strong>: a variable in an equation that needs to be solved for.<\/li>\r\n \t
    • equation<\/strong>: a mathematical statement that asserts the equivalence of two expressions.<\/li>\r\n \t
    • conditional equation<\/strong>: an equation that has a solution<\/li>\r\n \t
    • contradiction<\/strong>: an equation that has no solution<\/li>\r\n \t
    • identity<\/strong>: an equation that is always true<\/li>\r\n<\/ul>\r\n<\/div>\r\n

      What is an Equation?<\/h2>\r\nAn equation<\/strong><\/em> is a mathematical statement that asserts the equivalence of two expressions. For example, the assertion that \u201ctwo plus five equals seven\u201d is represented by the equation [latex]2 + 5 = 7[\/latex].\r\n\r\nIn most cases, an equation contains one or more variables. For example, the equation [latex]x + 3 = 5[\/latex], read \u201c[latex]x[\/latex]\u00a0plus three equals five\u201d, asserts that the expression [latex]x+3[\/latex]\u00a0is equal to the value [latex]5[\/latex].\r\n\r\nIt is possible for equations to have more than one variable. For example, [latex]x + y + 7 = 13[\/latex] is an equation in two variables, while [latex]5x^2+y^2+9z^2=36[\/latex] is an equation in three variables.\r\n

      Translating quantitative relationships into equations<\/h3>\r\nOne of the best uses of equations is to represent quantitative relationships, from which quantitative problems may be simplified and solved easily. For example, given the quantitative relationship \"three more than twice a number is equal to 27\", we may represent the relationship with the equation [latex]2x+3=27[\/latex]. If we use the letter [latex]x[\/latex] to represent the unknown number, the expression [latex]2x[\/latex] represents twice the unknown number [latex]x[\/latex]. We add 3 to [latex]2x[\/latex] (e.g., [latex]2x+3[\/latex]) to show three more than the number [latex]2x[\/latex]. Since the result (i.e., three more than twice a number) is equal to 27, we may translate the relationship into the equation\u00a0[latex]2x+3=27[\/latex].\r\n\r\nNow, the quantitative relationship is represented by a neat equation. The equation then may be further simplified and solved for finding the value of the unknown number [latex]x[\/latex]. You will learn to simplify and solve an equation in the next chapter.\r\n\r\n<\/div>\r\n
      \r\n\r\n \r\n
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      Examples<\/h3>\r\nHalf of an unknown number [latex]x[\/latex] is five less than double of another unknown number [latex]y[\/latex]. Write an equation to represent the quantitative relationship between the two numbers [latex]x[\/latex]\u00a0and [latex]y[\/latex].\r\n

      Solution<\/h4>\r\nHalf of the number [latex]x[\/latex] may be represented by the expression [latex]\\dfrac{x}{2}[\/latex].\r\n\r\nDouble of the number [latex]y[\/latex] may be represented by the expression [latex]2y[\/latex]. Five less than double of the number [latex]y[\/latex] is to decrease 5 from [latex]2y[\/latex], which may be represented by the expression [latex]2y-5[\/latex].\r\n\r\nSince the two parts (half of a number, five less than double of another number) are equal, the relationship may be written as the equation [latex]\\dfrac{x}{2}=2y-5[\/latex].\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n
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      Try It<\/h3>\r\nThe difference between 100 and an unknown number (smaller than 100) is equal to 48 less than the product of three and the unknown number. \u00a0Write an equation to represent the quantitative relationship. Use the letter [latex]x[\/latex] to represent the unknown number.\r\n\r\n[reveal-answer q=\"hjm074\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"hjm074\"]\r\n\r\n[latex]100-x=3x-48[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n \r\n

      Solving Equations<\/h3>\r\nWhen an equation contains a variable such as [latex]x[\/latex], this variable is considered an\u00a0unknown\u00a0<\/strong><\/em>value. In many cases, we can find the values for [latex]x[\/latex] that make the equation true. These values are called\u00a0solutions\u00a0<\/strong><\/em>of the equation.\r\n\r\nFor example, consider the equation we were talking about above: [latex]x + 3 =5[\/latex]. You have probably already guessed that the only possible value of [latex]x[\/latex] that makes the equation true is 2, because [latex]2 + 3 = 5[\/latex]. We use an equals sign to show that we know the value of a given variable. In this case, [latex]x=2[\/latex] is the only solution of the equation[latex]x + 3 =5[\/latex].\r\n\r\nThe values of the variables that make an equation true are called the\u00a0solutions\u00a0<\/em>of the equation. In turn, solving an equation<\/em> means determining what values for the variables make the equation a true statement.\r\n\r\nThe equation above was fairly straightforward; it was easy for us to identify the solution as [latex]x = 2[\/latex]. However, it becomes useful to have a process for finding solutions for unknowns as problems become more complex.\r\n

      Verifying Solutions<\/h3>\r\nIf a number is found as a solution of an equation, then substituting that number back into the equation in place of the variable should make the equation true. Thus, we can easily check whether a number is a genuine solution to a given equation.\r\n\r\nFor example, let\u2019s examine whether [latex]x=3[\/latex] is a solution to the equation\u00a0 [latex]2x + 31 = 37[\/latex].\r\n\r\nSubstituting 3 for [latex]x[\/latex], we have:\r\n

      [latex]2x + 31 = 37 \\\\ 2\\color{blue}{(3)} + 31 = 37 \\\\ 6 + 31 = 37 \\\\ 37 = 37[\/latex]<\/p>\r\nThis equality is a true statement. Therefore, we can conclude that [latex]x = 3[\/latex] is, in fact, a solution of the equation [latex]2x+31=37[\/latex].\r\n

      \r\n

      Examples<\/h3>\r\nDetermine whether or not [latex]x=-2[\/latex] is a solution of the following equations:\r\n\r\n1. [latex]3x+7=1[\/latex]\r\n\r\n2. [latex]-3x^2-x+10=0[\/latex]\r\n\r\n3. [latex]\\sqrt{x^2}=x[\/latex]\r\n

      Solution<\/h4>\r\nReplace [latex]x[\/latex] i each equation with [latex]-2[\/latex] and check if the equation is true.\r\n\r\n1. [latex]3x+7=1 \\\\ 3\\color{blue}{(-2)}+7=1 \\\\ -6+7=1 \\\\ 1=1[\/latex] TRUE\u00a0[latex]x=-2[\/latex] is a solution.\r\n\r\n2. [latex]-3x^2-x+10=0 \\\\ -3(\\color{blue}{(-2)}^2-\\color{blue}{(-2)}+10=0 \\\\ -3\\cdot 4 + 2 + 10 = 0 \\\\ -12 + 2 + 10 = 0 \\\\ 0 = 0[\/latex] TRUE\u00a0[latex]x=-2[\/latex] is a solution.\r\n\r\n3. [latex]\\sqrt{x^2}=x \\\\ \\sqrt{\\color{blue}{(-2)}^2}=\\color{blue}{(-2)} \\\\ \\sqrt{4}=-2 \\\\ 2=-2[\/latex] FALSE\u00a0[latex]x=-2[\/latex] is NOT a solution.\r\n\r\n<\/div>\r\n
      \r\n

      Try It<\/h3>\r\nDetermine whether or not [latex]x=3[\/latex] is a solution of the following equations:\r\n\r\n1. [latex]-2x+5=-1[\/latex]\r\n\r\n2. [latex]-2x^2+4x+30=0[\/latex]\r\n\r\n3. [latex]\\sqrt{4x^2}=2x[\/latex]\r\n\r\n[reveal-answer q=\"hjm074\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"hjm074\"]\r\n
        \r\n \t
      1. [latex]x=3[\/latex] is a solution of\u00a0[latex]-2x+5=-1[\/latex]<\/li>\r\n \t
      2. [latex]x=3[\/latex] is NOT a solution of\u00a0[latex]-2x^2+4x-6=0[\/latex]<\/li>\r\n \t
      3. [latex]x=3[\/latex] is a solution of\u00a0[latex]\\sqrt{4x^2}=2x[\/latex]<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n
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        Examples<\/h3>\r\nDetermine whether the pair of values [latex]x=1 \\text{ and }y=-2[\/latex] is a solution of the equation.\r\n\r\n1. [latex]4x+y=2[\/latex]\r\n\r\n2. [latex]x^2 + y^2=-3[\/latex]\r\n

        Solution<\/h4>\r\nReplace [latex]x[\/latex] with [\/altex]1[\/latex] and [latex]y[\/latex] with [latex]-2[\/latex].\r\n\r\n1. [latex]4x+y=2 \\\\ 4\\color{blue}{(1)}+\\color{blue}{(-2)}=2 \\\\ 4 + (-2) = 2 \\\\ 2 = 2[\/latex]\u00a0 TRUE.\u00a0[latex]x=1,\\,y=-2[\/latex] is a solution of the equation.\r\n\r\n \r\n\r\n2.[latex]x^2 + y^2 = -3 \\\\\\color{blue}{(1)}^2 +\\color{blue}{(-2)}^2 = -3 \\\\ 1 + 4 = 3 \\\\ 5 = 3 [\/latex] \u00a0FALSE.\u00a0[latex]x=1,\\,y=-2[\/latex] is NOT a solution of the equation.\r\n\r\n<\/div>\r\n
        \r\n

        Try It<\/h3>\r\nDetermine whether the pair of values [latex]x=2 \\text{ and }y=-3[\/latex] is a solution of the equation.\r\n\r\n1. [latex]x-y=-1[\/latex]\r\n\r\n2. [latex]x^2 - y^2=-5[\/latex]\r\n\r\n[reveal-answer q=\"hjm372\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"hjm372\"]\r\n
          \r\n \t
        1. [latex]x=2,\\,y=-3[\/latex] is a NOT solution of the equation.<\/li>\r\n \t
        2. [latex]x=2,\\,y=-3[\/latex] is a solution of the equation.<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n \r\n

          Classes of Equations<\/h2>\r\nEquations can be broadly classified into three categories:\r\n
            \r\n \t
          1. Conditional equations<\/li>\r\n \t
          2. Contradictions<\/li>\r\n \t
          3. Identities<\/li>\r\n<\/ol>\r\n \r\n\r\nLet's take a closer look at equations in each of these categories.\r\n\r\n<\/div>\r\n
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            Conditional Equations<\/h2>\r\n

            Case 1: Exactly One Solution<\/h3>\r\nThe one-variable equation [latex]x+2=5[\/latex] has only one solution [latex]x=3[\/latex] because the number 3 is the only value that satisfies the equation (makes the left side equal to the right side). The truth of the equation depends upon the value that is put in for [latex]x[\/latex]. In other words, the truth of the equation is conditional<\/em> on the value of\u00a0[latex]x[\/latex].\r\n

            Case 2: Infinitely Many Solutions (with a condition on the values)<\/h3>\r\nConsider the equation [latex]x+y=2[\/latex]. There are infinite number of solution pairs (x and y) for the equation.\r\n\r\nFor example,\r\n

            [latex]1+1=2[\/latex]\r\n[latex]2+0=2[\/latex]\r\n[latex]4+(\u20132)=2[\/latex]\r\n[latex]\\frac{1}{2}+\\frac{3}{2}=2[\/latex]<\/p>\r\nAny pair of two real numbers with a sum of 2 is a solution of this equation. Even though there are infinitely many solutions, the equation is CONDITIONAL because only those pairs that sum to 2 are solutions of the equation. All other pairs whose sum is not 2 cannot be solutions. A conditional equation means the solution of the equation is constrained to specific value(s).\r\n

            Contradictions<\/h2>\r\n

            No Solution:<\/h3>\r\nConsider the equation [latex]x^2=\u20134[\/latex]. There is no solution for this equation because the square of any real number is never negative.\r\n

            [latex](+)^2 = +[\/latex]\r\n[latex](\u2013)^2= +[\/latex]<\/p>\r\nTherefore, the equation [latex]x^2=\u20134[\/latex] is a contradiction\u00a0<\/strong><\/em>as it has no real solution.\r\n\r\nSimilarly, the equation [latex]|\\,x\\, |=\u20134[\/latex] is also a contradiction because the absolute value of any number is never negative.\r\n

            Identity<\/h2>\r\n

            Infinitely Many Solutions (with no condition on the values):<\/h3>\r\nAn identity<\/strong><\/em> is an equation where any value can be a solution of the equation.\r\n\r\nFor example, the equation\u00a0[latex]x=x[\/latex] is an identity because the left side of the equation is ALWAYS equal to the right side of the equation, regardless of the value of [latex]x[\/latex]. The left side and the right side are identical. Any value for the variable\u00a0[latex]x[\/latex] will be a solution because the left side is always equal to the right side no matter what value we plug in for the variable [latex]x[\/latex].\r\n
            \r\n

            Examples<\/h3>\r\nClassify each equation as conditional, a contradiction, or an identity, and identify how many solutions there are.\r\n\r\n1. [latex]5x-7=3[\/latex]\r\n\r\n2. [latex]\\sqrt{x-6}=-10[\/latex]\r\n\r\n3. [latex]x^4+5x^2+12=0[\/latex]\r\n\r\n4. [latex]4(3x-5)=12x-20[\/latex]\r\n

            Solution<\/h4>\r\n1. [latex]5x-7=3[\/latex] is a conditional equation<\/strong> because only [latex]x=2[\/latex] is a solution. One solution.\r\n\r\n2. [latex]\\sqrt{x-6}=-10[\/latex] is a\u00a0contradiction<\/strong> because the square root of any number cannot be negative. No solutions.\r\n\r\n3. [latex]x^4+5x^2+12=0[\/latex] is a\u00a0contradiction<\/strong> because each term is positive and the sum of the terms cannot equal zero. No solutions.\r\n\r\n4. [latex]4(3x-5)=12x-20[\/latex] is an\u00a0identity<\/strong> because this is an example of the distributive property that is true for all values of\u00a0[latex]x[\/latex]. Infinite solutions.\r\n\r\n<\/div>\r\n
            \r\n

            Try It<\/h3>\r\nClassify each equation as conditional, a contradiction, or an identity, and identify how many solutions there are.\r\n\r\n1. [latex]-2x + 2 = 0[\/latex]\r\n\r\n2. [latex]-\\sqrt{x} = -8[\/latex]\r\n\r\n3. [latex]\\large | \\normalsize -3x \\large | \\normalsize = -6[\/latex]\r\n\r\n4. [latex]2x - y= 4[\/latex]\r\n\r\n[reveal-answer q=\"539107\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"539107\"]\r\n
              \r\n \t
            1. Conditional. One solution. [latex]\\left ( x = 1 \\right )[\/latex]<\/li>\r\n \t
            2. Conditional. One solution. [latex]\\left ( x = 64 \\right )[\/latex]<\/li>\r\n \t
            3. Contradiction. No solutions. (absolute values cannot be negative)<\/li>\r\n \t
            4. Conditional. Infinite solutions.<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n \r\n\r\n \r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<\/div>\r\n<\/div>","rendered":"
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              Learning Objectives<\/h3>\n
                \n
              • Explain what an equation in one variable represents.<\/li>\n
              • Determine if a given value for a variable is a solution of an equation.<\/li>\n
              • Classify an equation as conditional, a contradiction or an identity.<\/li>\n<\/ul>\n<\/div>\n
                \n

                Key words<\/h3>\n
                  \n
                • solution<\/strong>: a value that can be substituted for a variable to make an equation true.<\/li>\n
                • unknown<\/strong>: a variable in an equation that needs to be solved for.<\/li>\n
                • equation<\/strong>: a mathematical statement that asserts the equivalence of two expressions.<\/li>\n
                • conditional equation<\/strong>: an equation that has a solution<\/li>\n
                • contradiction<\/strong>: an equation that has no solution<\/li>\n
                • identity<\/strong>: an equation that is always true<\/li>\n<\/ul>\n<\/div>\n

                  What is an Equation?<\/h2>\n

                  An equation<\/strong><\/em> is a mathematical statement that asserts the equivalence of two expressions. For example, the assertion that \u201ctwo plus five equals seven\u201d is represented by the equation [latex]2 + 5 = 7[\/latex].<\/p>\n

                  In most cases, an equation contains one or more variables. For example, the equation [latex]x + 3 = 5[\/latex], read \u201c[latex]x[\/latex]\u00a0plus three equals five\u201d, asserts that the expression [latex]x+3[\/latex]\u00a0is equal to the value [latex]5[\/latex].<\/p>\n

                  It is possible for equations to have more than one variable. For example, [latex]x + y + 7 = 13[\/latex] is an equation in two variables, while [latex]5x^2+y^2+9z^2=36[\/latex] is an equation in three variables.<\/p>\n

                  Translating quantitative relationships into equations<\/h3>\n

                  One of the best uses of equations is to represent quantitative relationships, from which quantitative problems may be simplified and solved easily. For example, given the quantitative relationship “three more than twice a number is equal to 27”, we may represent the relationship with the equation [latex]2x+3=27[\/latex]. If we use the letter [latex]x[\/latex] to represent the unknown number, the expression [latex]2x[\/latex] represents twice the unknown number [latex]x[\/latex]. We add 3 to [latex]2x[\/latex] (e.g., [latex]2x+3[\/latex]) to show three more than the number [latex]2x[\/latex]. Since the result (i.e., three more than twice a number) is equal to 27, we may translate the relationship into the equation\u00a0[latex]2x+3=27[\/latex].<\/p>\n

                  Now, the quantitative relationship is represented by a neat equation. The equation then may be further simplified and solved for finding the value of the unknown number [latex]x[\/latex]. You will learn to simplify and solve an equation in the next chapter.<\/p>\n<\/div>\n

                  \n

                   <\/p>\n

                  \n

                  Examples<\/h3>\n

                  Half of an unknown number [latex]x[\/latex] is five less than double of another unknown number [latex]y[\/latex]. Write an equation to represent the quantitative relationship between the two numbers [latex]x[\/latex]\u00a0and [latex]y[\/latex].<\/p>\n

                  Solution<\/h4>\n

                  Half of the number [latex]x[\/latex] may be represented by the expression [latex]\\dfrac{x}{2}[\/latex].<\/p>\n

                  Double of the number [latex]y[\/latex] may be represented by the expression [latex]2y[\/latex]. Five less than double of the number [latex]y[\/latex] is to decrease 5 from [latex]2y[\/latex], which may be represented by the expression [latex]2y-5[\/latex].<\/p>\n

                  Since the two parts (half of a number, five less than double of another number) are equal, the relationship may be written as the equation [latex]\\dfrac{x}{2}=2y-5[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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                  Try It<\/h3>\n

                  The difference between 100 and an unknown number (smaller than 100) is equal to 48 less than the product of three and the unknown number. \u00a0Write an equation to represent the quantitative relationship. Use the letter [latex]x[\/latex] to represent the unknown number.<\/p>\n

                  Show Answer<\/span><\/p>\n
                  \n

                  [latex]100-x=3x-48[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n

                   <\/p>\n

                  Solving Equations<\/h3>\n

                  When an equation contains a variable such as [latex]x[\/latex], this variable is considered an\u00a0unknown\u00a0<\/strong><\/em>value. In many cases, we can find the values for [latex]x[\/latex] that make the equation true. These values are called\u00a0solutions\u00a0<\/strong><\/em>of the equation.<\/p>\n

                  For example, consider the equation we were talking about above: [latex]x + 3 =5[\/latex]. You have probably already guessed that the only possible value of [latex]x[\/latex] that makes the equation true is 2, because [latex]2 + 3 = 5[\/latex]. We use an equals sign to show that we know the value of a given variable. In this case, [latex]x=2[\/latex] is the only solution of the equation[latex]x + 3 =5[\/latex].<\/p>\n

                  The values of the variables that make an equation true are called the\u00a0solutions\u00a0<\/em>of the equation. In turn, solving an equation<\/em> means determining what values for the variables make the equation a true statement.<\/p>\n

                  The equation above was fairly straightforward; it was easy for us to identify the solution as [latex]x = 2[\/latex]. However, it becomes useful to have a process for finding solutions for unknowns as problems become more complex.<\/p>\n

                  Verifying Solutions<\/h3>\n

                  If a number is found as a solution of an equation, then substituting that number back into the equation in place of the variable should make the equation true. Thus, we can easily check whether a number is a genuine solution to a given equation.<\/p>\n

                  For example, let\u2019s examine whether [latex]x=3[\/latex] is a solution to the equation\u00a0 [latex]2x + 31 = 37[\/latex].<\/p>\n

                  Substituting 3 for [latex]x[\/latex], we have:<\/p>\n

                  [latex]2x + 31 = 37 \\\\ 2\\color{blue}{(3)} + 31 = 37 \\\\ 6 + 31 = 37 \\\\ 37 = 37[\/latex]<\/p>\n

                  This equality is a true statement. Therefore, we can conclude that [latex]x = 3[\/latex] is, in fact, a solution of the equation [latex]2x+31=37[\/latex].<\/p>\n

                  \n

                  Examples<\/h3>\n

                  Determine whether or not [latex]x=-2[\/latex] is a solution of the following equations:<\/p>\n

                  1. [latex]3x+7=1[\/latex]<\/p>\n

                  2. [latex]-3x^2-x+10=0[\/latex]<\/p>\n

                  3. [latex]\\sqrt{x^2}=x[\/latex]<\/p>\n

                  Solution<\/h4>\n

                  Replace [latex]x[\/latex] i each equation with [latex]-2[\/latex] and check if the equation is true.<\/p>\n

                  1. [latex]3x+7=1 \\\\ 3\\color{blue}{(-2)}+7=1 \\\\ -6+7=1 \\\\ 1=1[\/latex] TRUE\u00a0[latex]x=-2[\/latex] is a solution.<\/p>\n

                  2. [latex]-3x^2-x+10=0 \\\\ -3(\\color{blue}{(-2)}^2-\\color{blue}{(-2)}+10=0 \\\\ -3\\cdot 4 + 2 + 10 = 0 \\\\ -12 + 2 + 10 = 0 \\\\ 0 = 0[\/latex] TRUE\u00a0[latex]x=-2[\/latex] is a solution.<\/p>\n

                  3. [latex]\\sqrt{x^2}=x \\\\ \\sqrt{\\color{blue}{(-2)}^2}=\\color{blue}{(-2)} \\\\ \\sqrt{4}=-2 \\\\ 2=-2[\/latex] FALSE\u00a0[latex]x=-2[\/latex] is NOT a solution.<\/p>\n<\/div>\n

                  \n

                  Try It<\/h3>\n

                  Determine whether or not [latex]x=3[\/latex] is a solution of the following equations:<\/p>\n

                  1. [latex]-2x+5=-1[\/latex]<\/p>\n

                  2. [latex]-2x^2+4x+30=0[\/latex]<\/p>\n

                  3. [latex]\\sqrt{4x^2}=2x[\/latex]<\/p>\n

                  Show Answer<\/span><\/p>\n
                  \n
                    \n
                  1. [latex]x=3[\/latex] is a solution of\u00a0[latex]-2x+5=-1[\/latex]<\/li>\n
                  2. [latex]x=3[\/latex] is NOT a solution of\u00a0[latex]-2x^2+4x-6=0[\/latex]<\/li>\n
                  3. [latex]x=3[\/latex] is a solution of\u00a0[latex]\\sqrt{4x^2}=2x[\/latex]<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
                    \n
                    \n
                    \n
                    \n

                    Examples<\/h3>\n

                    Determine whether the pair of values [latex]x=1 \\text{ and }y=-2[\/latex] is a solution of the equation.<\/p>\n

                    1. [latex]4x+y=2[\/latex]<\/p>\n

                    2. [latex]x^2 + y^2=-3[\/latex]<\/p>\n

                    Solution<\/h4>\n

                    Replace [latex]x[\/latex] with [\/altex]1[\/latex] and [latex]y[\/latex] with [latex]-2[\/latex].<\/p>\n

                    1. [latex]4x+y=2 \\\\ 4\\color{blue}{(1)}+\\color{blue}{(-2)}=2 \\\\ 4 + (-2) = 2 \\\\ 2 = 2[\/latex]\u00a0 TRUE.\u00a0[latex]x=1,\\,y=-2[\/latex] is a solution of the equation.<\/p>\n

                     <\/p>\n

                    2.[latex]x^2 + y^2 = -3 \\\\\\color{blue}{(1)}^2 +\\color{blue}{(-2)}^2 = -3 \\\\ 1 + 4 = 3 \\\\ 5 = 3[\/latex] \u00a0FALSE.\u00a0[latex]x=1,\\,y=-2[\/latex] is NOT a solution of the equation.<\/p>\n<\/div>\n

                    \n

                    Try It<\/h3>\n

                    Determine whether the pair of values [latex]x=2 \\text{ and }y=-3[\/latex] is a solution of the equation.<\/p>\n

                    1. [latex]x-y=-1[\/latex]<\/p>\n

                    2. [latex]x^2 - y^2=-5[\/latex]<\/p>\n

                    Show Answer<\/span><\/p>\n
                    \n
                      \n
                    1. [latex]x=2,\\,y=-3[\/latex] is a NOT solution of the equation.<\/li>\n
                    2. [latex]x=2,\\,y=-3[\/latex] is a solution of the equation.<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n

                       <\/p>\n

                      Classes of Equations<\/h2>\n

                      Equations can be broadly classified into three categories:<\/p>\n

                        \n
                      1. Conditional equations<\/li>\n
                      2. Contradictions<\/li>\n
                      3. Identities<\/li>\n<\/ol>\n

                         <\/p>\n

                        Let’s take a closer look at equations in each of these categories.<\/p>\n<\/div>\n

                        \n
                        \n

                        Conditional Equations<\/h2>\n

                        Case 1: Exactly One Solution<\/h3>\n

                        The one-variable equation [latex]x+2=5[\/latex] has only one solution [latex]x=3[\/latex] because the number 3 is the only value that satisfies the equation (makes the left side equal to the right side). The truth of the equation depends upon the value that is put in for [latex]x[\/latex]. In other words, the truth of the equation is conditional<\/em> on the value of\u00a0[latex]x[\/latex].<\/p>\n

                        Case 2: Infinitely Many Solutions (with a condition on the values)<\/h3>\n

                        Consider the equation [latex]x+y=2[\/latex]. There are infinite number of solution pairs (x and y) for the equation.<\/p>\n

                        For example,<\/p>\n

                        [latex]1+1=2[\/latex]
                        \n[latex]2+0=2[\/latex]
                        \n[latex]4+(\u20132)=2[\/latex]
                        \n[latex]\\frac{1}{2}+\\frac{3}{2}=2[\/latex]<\/p>\n

                        Any pair of two real numbers with a sum of 2 is a solution of this equation. Even though there are infinitely many solutions, the equation is CONDITIONAL because only those pairs that sum to 2 are solutions of the equation. All other pairs whose sum is not 2 cannot be solutions. A conditional equation means the solution of the equation is constrained to specific value(s).<\/p>\n

                        Contradictions<\/h2>\n

                        No Solution:<\/h3>\n

                        Consider the equation [latex]x^2=\u20134[\/latex]. There is no solution for this equation because the square of any real number is never negative.<\/p>\n

                        [latex](+)^2 = +[\/latex]
                        \n[latex](\u2013)^2= +[\/latex]<\/p>\n

                        Therefore, the equation [latex]x^2=\u20134[\/latex] is a contradiction\u00a0<\/strong><\/em>as it has no real solution.<\/p>\n

                        Similarly, the equation [latex]|\\,x\\, |=\u20134[\/latex] is also a contradiction because the absolute value of any number is never negative.<\/p>\n

                        Identity<\/h2>\n

                        Infinitely Many Solutions (with no condition on the values):<\/h3>\n

                        An identity<\/strong><\/em> is an equation where any value can be a solution of the equation.<\/p>\n

                        For example, the equation\u00a0[latex]x=x[\/latex] is an identity because the left side of the equation is ALWAYS equal to the right side of the equation, regardless of the value of [latex]x[\/latex]. The left side and the right side are identical. Any value for the variable\u00a0[latex]x[\/latex] will be a solution because the left side is always equal to the right side no matter what value we plug in for the variable [latex]x[\/latex].<\/p>\n

                        \n

                        Examples<\/h3>\n

                        Classify each equation as conditional, a contradiction, or an identity, and identify how many solutions there are.<\/p>\n

                        1. [latex]5x-7=3[\/latex]<\/p>\n

                        2. [latex]\\sqrt{x-6}=-10[\/latex]<\/p>\n

                        3. [latex]x^4+5x^2+12=0[\/latex]<\/p>\n

                        4. [latex]4(3x-5)=12x-20[\/latex]<\/p>\n

                        Solution<\/h4>\n

                        1. [latex]5x-7=3[\/latex] is a conditional equation<\/strong> because only [latex]x=2[\/latex] is a solution. One solution.<\/p>\n

                        2. [latex]\\sqrt{x-6}=-10[\/latex] is a\u00a0contradiction<\/strong> because the square root of any number cannot be negative. No solutions.<\/p>\n

                        3. [latex]x^4+5x^2+12=0[\/latex] is a\u00a0contradiction<\/strong> because each term is positive and the sum of the terms cannot equal zero. No solutions.<\/p>\n

                        4. [latex]4(3x-5)=12x-20[\/latex] is an\u00a0identity<\/strong> because this is an example of the distributive property that is true for all values of\u00a0[latex]x[\/latex]. Infinite solutions.<\/p>\n<\/div>\n

                        \n

                        Try It<\/h3>\n

                        Classify each equation as conditional, a contradiction, or an identity, and identify how many solutions there are.<\/p>\n

                        1. [latex]-2x + 2 = 0[\/latex]<\/p>\n

                        2. [latex]-\\sqrt{x} = -8[\/latex]<\/p>\n

                        3. [latex]\\large | \\normalsize -3x \\large | \\normalsize = -6[\/latex]<\/p>\n

                        4. [latex]2x - y= 4[\/latex]<\/p>\n

                        Show Answer<\/span><\/p>\n
                        \n
                          \n
                        1. Conditional. One solution. [latex]\\left ( x = 1 \\right )[\/latex]<\/li>\n
                        2. Conditional. One solution. [latex]\\left ( x = 64 \\right )[\/latex]<\/li>\n
                        3. Contradiction. No solutions. (absolute values cannot be negative)<\/li>\n
                        4. Conditional. Infinite solutions.<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n

                           <\/p>\n

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