NEW CHAPTER 5 Variables, Expressions and Equations

5.4 Introduction to Equations

Learning Objectives

  • Explain what an equation in one variable represents.
  • Determine if a given value for a variable is a solution of an equation.
  • Classify an equation as conditional, a contradiction or an identity.

Key words

  • solution: a value that can be substituted for a variable to make an equation true.
  • unknown: a variable in an equation that needs to be solved for.
  • equation: a mathematical statement that asserts the equivalence of two expressions.
  • conditional equation: an equation that has a solution
  • contradiction: an equation that has no solution
  • identity: an equation that is always true

What is an Equation?

An equation is a mathematical statement that asserts the equivalence of two expressions. For example, the assertion that “two plus five equals seven” is represented by the equation [latex]2 + 5 = 7[/latex].

In most cases, an equation contains one or more variables. For example, the equation [latex]x + 3 = 5[/latex], read “[latex]x[/latex] plus three equals five”, asserts that the expression [latex]x+3[/latex] is equal to the value [latex]5[/latex].

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.

Translating quantitative relationships into equations

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 [latex]2x+3=27[/latex].

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.

 

Examples

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] and [latex]y[/latex].

Solution

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

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].

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].

Try It

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.  Write an equation to represent the quantitative relationship. Use the letter [latex]x[/latex] to represent the unknown number.

 

Solving Equations

When an equation contains a variable such as [latex]x[/latex], this variable is considered an unknown value. In many cases, we can find the values for [latex]x[/latex] that make the equation true. These values are called solutions of the equation.

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].

The values of the variables that make an equation true are called the solutions of the equation. In turn, solving an equation means determining what values for the variables make the equation a true statement.

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.

Verifying Solutions

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.

For example, let’s examine whether [latex]x=3[/latex] is a solution to the equation  [latex]2x + 31 = 37[/latex].

Substituting 3 for [latex]x[/latex], we have:

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

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].

Examples

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

1. [latex]3x+7=1[/latex]

2. [latex]-3x^2-x+10=0[/latex]

3. [latex]\sqrt{x^2}=x[/latex]

Solution

Replace [latex]x[/latex] i each equation with [latex]-2[/latex] and check if the equation is true.

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

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 [latex]x=-2[/latex] is a solution.

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

Try It

Determine whether or not [latex]x=3[/latex] is a solution of the following equations:

1. [latex]-2x+5=-1[/latex]

2. [latex]-2x^2+4x+30=0[/latex]

3. [latex]\sqrt{4x^2}=2x[/latex]

Examples

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

1. [latex]4x+y=2[/latex]

2. [latex]x^2 + y^2=-3[/latex]

Solution

Replace [latex]x[/latex] with [/altex]1[/latex] and [latex]y[/latex] with [latex]-2[/latex].

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

 

2.[latex]x^2 + y^2 = -3 \\\color{blue}{(1)}^2 +\color{blue}{(-2)}^2 = -3 \\ 1 + 4 = 3 \\ 5 = 3[/latex]  FALSE. [latex]x=1,\,y=-2[/latex] is NOT a solution of the equation.

Try It

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

1. [latex]x-y=-1[/latex]

2. [latex]x^2 - y^2=-5[/latex]

 

Classes of Equations

Equations can be broadly classified into three categories:

  1. Conditional equations
  2. Contradictions
  3. Identities

 

Let’s take a closer look at equations in each of these categories.

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