{"id":632,"date":"2021-08-30T20:43:13","date_gmt":"2021-08-30T20:43:13","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/?post_type=chapter&p=632"},"modified":"2024-01-23T23:10:23","modified_gmt":"2024-01-23T23:10:23","slug":"3-3-equations-and-solutions","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/chapter\/3-3-equations-and-solutions\/","title":{"raw":"3.3: Introduction to Equations","rendered":"3.3: Introduction to Equations"},"content":{"raw":"
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Learning Objectives<\/h3>\r\n
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  • 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
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    Key words<\/h3>\r\n
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    • 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

      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

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      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
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      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
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      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
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        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
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        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
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          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
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            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
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            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
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            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