Linear Equations in One Variable

Learning Objectives

In this section, you will:

• Solve equations in one variable algebraically.
• Solve a rational equation.
• Find a linear equation.
• Given the equations of two lines, determine whether their graphs are parallel or perpendicular.
• Write the equation of a line parallel or perpendicular to a given line.

Caroline is a full-time college student planning a spring break vacation. To earn enough money for the trip, she has taken a part-time job at the local bank that pays $15.00/hr, and she opened a savings account with an initial deposit of $400 on January 15. She arranged for direct deposit of her payroll checks. If spring break begins March 20 and the trip will cost approximately $2,500, how many hours will she have to work to earn enough to pay for her vacation? If she can only work 4 hours per day, how many days per week will she have to work? How many weeks will it take? In this section, we will investigate problems like this and others, which generate graphs like the line in Figure 1.

Figure 1

Solving Linear Equations in One Variable

A linear equation is an equation of a straight line, written in one variable. The only power of the variable is 1. Linear equations in one variable may take the form ax+b=0𝑎𝑥+𝑏=0 and are solved using basic algebraic operations.

We begin by classifying linear equations in one variable as one of three types: identity, conditional, or inconsistent. An identity equation is true for all values of the variable. Here is an example of an identity equation.

3x=2x+x3𝑥=2𝑥+𝑥

The solution set consists of all values that make the equation true. For this equation, the solution set is all real numbers because any real number substituted for x𝑥 will make the equation true.

A conditional equation is true for only some values of the variable. For example, if we are to solve the equation 5x+2=3x−6,5𝑥+2=3𝑥−6, we have the following:

5x+22xx===3x−6−8−45𝑥+2=3𝑥−62𝑥=−8𝑥=−4

The solution set consists of one number: {−4}.{−4}. It is the only solution and, therefore, we have solved a conditional equation.

An inconsistent equation results in a false statement. For example, if we are to solve 5x−15=5(x−4),5𝑥−15=5(𝑥−4), we have the following:

5x−155x−15−5x−15==≠5x−205x−20−5x−20Subtract 5xfrom both sides.False statement5𝑥−15=5𝑥−205𝑥−15−5𝑥=5𝑥−20−5𝑥Subtract 5𝑥from both sides.−15≠−20False statement

Indeed, −15≠−20.−15≠−20. There is no solution because this is an inconsistent equation.

Solving linear equations in one variable involves the fundamental properties of equality and basic algebraic operations. A brief review of those operations follows.

LINEAR EQUATION IN ONE VARIABLE

A linear equation in one variable can be written in the form

ax+b=0𝑎𝑥+𝑏=0

where a and b are real numbers, a≠0.𝑎≠0.

HOW TO

Given a linear equation in one variable, use algebra to solve it.

The following steps are used to manipulate an equation and isolate the unknown variable, so that the last line reads x=_________,𝑥=_________, if x is the unknown. There is no set order, as the steps used depend on what is given:

1. We may add, subtract, multiply, or divide an equation by a number or an expression as long as we do the same thing to both sides of the equal sign. Note that we cannot divide by zero.
2. Apply the distributive property as needed: a(b+c)=ab+ac.𝑎(𝑏+𝑐)=𝑎𝑏+𝑎𝑐.
3. Isolate the variable on one side of the equation.
4. When the variable is multiplied by a coefficient in the final stage, multiply both sides of the equation by the reciprocal of the coefficient.

EXAMPLE 1

Solving an Equation in One Variable

Solve the following equation: 2x+7=19.2𝑥+7=19.

TRY IT #1

Solve the linear equation in one variable: 2x+1=−9.2𝑥+1=−9.

EXAMPLE 2

Solving an Equation Algebraically When the Variable Appears on Both Sides

Solve the following equation: 4(x−3)+12=15−5(x+6).4(𝑥−3)+12=15−5(𝑥+6).

Analysis

This problem requires the distributive property to be applied twice, and then the properties of algebra are used to reach the final line, x=−53.𝑥=−53.

TRY IT #2

Solve the equation in one variable: −2(3x−1)+x=14−x.−2(3𝑥−1)+𝑥=14−𝑥.

Solving a Rational Equation

In this section, we look at rational equations that, after some manipulation, result in a linear equation. If an equation contains at least one rational expression, it is a considered a rational equation.

Recall that a rational number is the ratio of two numbers, such as 2323 or 72.72. A rational expression is the ratio, or quotient, of two polynomials. Here are three examples.

x+1×2−4,1x−3,or4x2+x−2𝑥+1𝑥2−4,1𝑥−3,or4𝑥2+𝑥−2

Rational equations have a variable in the denominator in at least one of the terms. Our goal is to perform algebraic operations so that the variables appear in the numerator. In fact, we will eliminate all denominators by multiplying both sides of the equation by the least common denominator (LCD).

Finding the LCD is identifying an expression that contains the highest power of all of the factors in all of the denominators. We do this because when the equation is multiplied by the LCD, the common factors in the LCD and in each denominator will equal one and will cancel out.

EXAMPLE 3

Solving a Rational Equation

Solve the rational equation: 72x−53x=223.72𝑥−53𝑥=223.

A common mistake made when solving rational equations involves finding the LCD when one of the denominators is a binomial—two terms added or subtracted—such as (x+1).(𝑥+1). Always consider a binomial as an individual factor—the terms cannot be separated. For example, suppose a problem has three terms and the denominators are x,𝑥, x−1,𝑥−1, and 3x−3.3𝑥−3. First, factor all denominators. We then have x,𝑥, (x−1),(𝑥−1), and 3(x−1)3(𝑥−1) as the denominators. (Note the parentheses placed around the second denominator.) Only the last two denominators have a common factor of (x−1).(𝑥−1). The x𝑥 in the first denominator is separate from the x𝑥 in the (x−1)(𝑥−1) denominators. An effective way to remember this is to write factored and binomial denominators in parentheses, and consider each parentheses as a separate unit or a separate factor. The LCD in this instance is found by multiplying together the x,𝑥, one factor of (x−1),(𝑥−1), and the 3. Thus, the LCD is the following:

x(x−1)3=3x(x−1)𝑥(𝑥−1)3=3𝑥(𝑥−1)

So, both sides of the equation would be multiplied by 3x(x−1).3𝑥(𝑥−1). Leave the LCD in factored form, as this makes it easier to see how each denominator in the problem cancels out.

Another example is a problem with two denominators, such as x𝑥 and x2+2x.𝑥2+2𝑥. Once the second denominator is factored as x2+2x=x(x+2),𝑥2+2𝑥=𝑥(𝑥+2), there is a common factor of x in both denominators and the LCD is x(x+2).𝑥(𝑥+2).

Sometimes we have a rational equation in the form of a proportion; that is, when one fraction equals another fraction and there are no other terms in the equation.

ab=cd𝑎𝑏=𝑐𝑑

We can use another method of solving the equation without finding the LCD: cross-multiplication. We multiply terms by crossing over the equal sign.

Multiply a(d)𝑎(𝑑) and b(c),𝑏(𝑐), which results in ad=bc.𝑎𝑑=𝑏𝑐.

Any solution that makes a denominator in the original expression equal zero must be excluded from the possibilities.

RATIONAL EQUATIONS

A rational equation contains at least one rational expression where the variable appears in at least one of the denominators.

HOW TO

Given a rational equation, solve it.

1. Factor all denominators in the equation.
2. Find and exclude values that set each denominator equal to zero.
3. Find the LCD.
4. Multiply the whole equation by the LCD. If the LCD is correct, there will be no denominators left.
5. Solve the remaining equation.
6. Make sure to check solutions back in the original equations to avoid a solution producing zero in a denominator.

EXAMPLE 4

Solving a Rational Equation without Factoring

Solve the following rational equation:

2x−32=72×2𝑥−32=72𝑥

TRY IT #3

Solve the rational equation: 23x=14−16x.23𝑥=14−16𝑥.

EXAMPLE 5

Solving a Rational Equation by Factoring the Denominator

Solve the following rational equation: 1x=110−34x.1𝑥=110−34𝑥.

TRY IT #4

Solve the rational equation: −52x+34x=−74.−52𝑥+34𝑥=−74.

EXAMPLE 6

Solving Rational Equations with a Binomial in the Denominator

Solve the following rational equations and state the excluded values:

1. ⓐ 3x−6=5×3𝑥−6=5𝑥
2. ⓑ xx−3=5x−3−12𝑥𝑥−3=5𝑥−3−12
3. ⓒ xx−2=5x−2−12𝑥𝑥−2=5𝑥−2−12

TRY IT #5

Solve −32x+1=43x+1.−32𝑥+1=43𝑥+1. State the excluded values.

EXAMPLE 7

Solving a Rational Equation with Factored Denominators and Stating Excluded Values

Solve the rational equation after factoring the denominators: 2x+1−1x−1=2xx2−1.2𝑥+1−1𝑥−1=2𝑥𝑥2−1. State the excluded values.

TRY IT #6

Solve the rational equation: 2x−2+1x+1=1×2−x−2.2𝑥−2+1𝑥+1=1𝑥2−𝑥−2.

Finding a Linear Equation

Perhaps the most familiar form of a linear equation is the slope-intercept form, written as y=mx+b,𝑦=𝑚𝑥+𝑏, where m=slope𝑚=slope and b=y-intercept.𝑏=𝑦-intercept. Let us begin with the slope.

The Slope of a Line

The slope of a line refers to the ratio of the vertical change in y over the horizontal change in x between any two points on a line. It indicates the direction in which a line slants as well as its steepness. Slope is sometimes described as rise over run.

m=y2−y1x2−x1𝑚=𝑦2−𝑦1𝑥2−𝑥1

If the slope is positive, the line slants to the right. If the slope is negative, the line slants to the left. As the slope increases, the line becomes steeper. Some examples are shown in Figure 2. The lines indicate the following slopes: m=−3,𝑚=−3, m=2,𝑚=2, and m=13.𝑚=13.

Figure 2

THE SLOPE OF A LINE

The slope of a line, m, represents the change in y over the change in x. Given two points, (x1,y1)(𝑥1,𝑦1) and (x2,y2),(𝑥2,𝑦2), the following formula determines the slope of a line containing these points:

m=y2−y1x2−x1𝑚=𝑦2−𝑦1𝑥2−𝑥1

EXAMPLE 8

Finding the Slope of a Line Given Two Points

Find the slope of a line that passes through the points (2,−1)(2,−1) and (−5,3).(−5,3)

Analysis

It does not matter which point is called (x1,y1)(𝑥1,𝑦1) or (x2,y2).(𝑥2,𝑦2). As long as we are consistent with the order of the y terms and the order of the x terms in the numerator and denominator, the calculation will yield the same result.

TRY IT #7

Find the slope of the line that passes through the points (−2,6)(−2,6) and (1,4).(1,4).

EXAMPLE 9

Identifying the Slope and y-intercept of a Line Given an Equation

Identify the slope and y-intercept, given the equation y=−34x−4.𝑦=−34𝑥−4.

Analysis

The y-intercept is the point at which the line crosses the y-axis. On the y-axis, x=0.𝑥=0. We can always identify the y-intercept when the line is in slope-intercept form, as it will always equal b. Or, just substitute x=0𝑥=0 and solve for y.

The Point-Slope Formula

Given the slope and one point on a line, we can find the equation of the line using the point-slope formula.

y−y1=m(x−x1)𝑦−𝑦1=𝑚(𝑥−𝑥1)

This is an important formula, as it will be used in other areas of college algebra and often in calculus to find the equation of a tangent line. We need only one point and the slope of the line to use the formula. After substituting the slope and the coordinates of one point into the formula, we simplify it and write it in slope-intercept form.

THE POINT-SLOPE FORMULA

Given one point and the slope, the point-slope formula will lead to the equation of a line:

y−y1=m(x−x1)𝑦−𝑦1=𝑚(𝑥−𝑥1)

EXAMPLE 10

Finding the Equation of a Line Given the Slope and One Point

Write the equation of the line with slope m=−3𝑚=−3 and passing through the point (4,8).(4,8). Write the final equation in slope-intercept form.

Analysis

Note that any point on the line can be used to find the equation. If done correctly, the same final equation will be obtained.

TRY IT #8

Given m=4,𝑚=4, find the equation of the line in slope-intercept form passing through the point (2,5).(2,5).

EXAMPLE 11

Finding the Equation of a Line Passing Through Two Given Points

Find the equation of the line passing through the points (3,4)(3,4) and (0,−3).(0,−3). Write the final equation in slope-intercept form.

Analysis

To prove that either point can be used, let us use the second point (0,−3)(0,−3) and see if we get the same equation.

y−(−3)y+3y===73(x−0)73x73x−3𝑦−(−3)=73(𝑥−0)𝑦+3=73𝑥𝑦=73𝑥−3

We see that the same line will be obtained using either point. This makes sense because we used both points to calculate the slope.

Standard Form of a Line

Another way that we can represent the equation of a line is in standard form. Standard form is given as

Ax+By=C𝐴𝑥+𝐵𝑦=𝐶

where A,𝐴, B,𝐵, and C𝐶 are integers. The x- and y-terms are on one side of the equal sign and the constant term is on the other side.

EXAMPLE 12

Finding the Equation of a Line and Writing It in Standard Form

Find the equation of the line with m=−6𝑚=−6 and passing through the point (14,−2).(14,−2). Write the equation in standard form.

TRY IT #9

Find the equation of the line in standard form with slope m=−13𝑚=−13 and passing through the point (1,13).(1,13).

Vertical and Horizontal Lines

The equations of vertical and horizontal lines do not require any of the preceding formulas, although we can use the formulas to prove that the equations are correct. The equation of a vertical line is given as

x=c𝑥=𝑐

where c is a constant. The slope of a vertical line is undefined, and regardless of the y-value of any point on the line, the x-coordinate of the point will be c.

Suppose that we want to find the equation of a line containing the following points: (−3,−5),(−3,1),(−3,3),(−3,−5),(−3,1),(−3,3), and (−3,5).(−3,5). First, we will find the slope.

m=5−3−3−(−3)=20𝑚=5−3−3−(−3)=20

Zero in the denominator means that the slope is undefined and, therefore, we cannot use the point-slope formula. However, we can plot the points. Notice that all of the x-coordinates are the same and we find a vertical line through x=−3.𝑥=−3. See Figure 3.

The equation of a horizontal line is given as

y=c𝑦=𝑐

where c is a constant. The slope of a horizontal line is zero, and for any x-value of a point on the line, the y-coordinate will be c.

Suppose we want to find the equation of a line that contains the following set of points: (−2,−2),(0,−2),(3,−2),(−2,−2),(0,−2),(3,−2), and (5,−2).(5,−2). We can use the point-slope formula. First, we find the slope using any two points on the line.

m===−2−(−2)0−(−2)020𝑚=−2−(−2)0−(−2)=02=0

Use any point for (x1,y1)(𝑥1,𝑦1) in the formula, or use the y-intercept.

y−(−2)y+2y===0(x−3)0−2𝑦−(−2)=0(𝑥−3)𝑦+2=0𝑦=−2

The graph is a horizontal line through y=−2.𝑦=−2. Notice that all of the y-coordinates are the same. See Figure 3.

Figure 3 The line x = −3 is a vertical line. The line y = −2 is a horizontal line.

EXAMPLE 13

Finding the Equation of a Line Passing Through the Given Points

Find the equation of the line passing through the given points: (1,−3)(1,−3) and (1,4).(1,4)

TRY IT #10

Find the equation of the line passing through (−5,2)(−5,2) and (2,2).(2,2).

Determining Whether Graphs of Lines are Parallel or Perpendicular

Parallel lines have the same slope and different y-intercepts. Lines that are parallel to each other will never intersect. For example, Figure 4 shows the graphs of various lines with the same slope, m=2.𝑚=2.

Figure 4 Parallel lines

All of the lines shown in the graph are parallel because they have the same slope and different y-intercepts.

Lines that are perpendicular intersect to form a 90°90° -angle. The slope of one line is the negative reciprocal of the other. We can show that two lines are perpendicular if the product of the two slopes is −1:m1⋅m2=−1.−1:𝑚1⋅𝑚2=−1. For example, Figure 5 shows the graph of two perpendicular lines. One line has a slope of 3; the other line has a slope of −13.−13.

m1⋅m23⋅(−13)==−1−1𝑚1⋅𝑚2=−13⋅(−13)=−1

Figure 5 Perpendicular lines

EXAMPLE 14

Graphing Two Equations, and Determining Whether the Lines are Parallel, Perpendicular, or Neither

Graph the equations of the given lines, and state whether they are parallel, perpendicular, or neither: 3y=−4x+33𝑦=−4𝑥+3 and 3x−4y=8.3𝑥−4𝑦=8

TRY IT #11

Graph the two lines and determine whether they are parallel, perpendicular, or neither: 2y−x=102𝑦−𝑥=10 and 2y=x+4.2𝑦=𝑥+4.

Writing the Equations of Lines Parallel or Perpendicular to a Given Line

As we have learned, determining whether two lines are parallel or perpendicular is a matter of finding the slopes. To write the equation of a line parallel or perpendicular to another line, we follow the same principles as we do for finding the equation of any line. After finding the slope, use the point-slope formula to write the equation of the new line.

HOW TO

Given an equation for a line, write the equation of a line parallel or perpendicular to it.

1. Find the slope of the given line. The easiest way to do this is to write the equation in slope-intercept form.
2. Use the slope and the given point with the point-slope formula.
3. Simplify the line to slope-intercept form and compare the equation to the given line.

EXAMPLE 15

Writing the Equation of a Line Parallel to a Given Line Passing Through a Given Point

Write the equation of line parallel to a 5x+3y=15𝑥+3𝑦=1 and passing through the point (3,5).(3,5).

TRY IT #12

Find the equation of the line parallel to 5x=7+y5𝑥=7+𝑦 and passing through the point (−1,−2).(−1,−2).

EXAMPLE 16

Finding the Equation of a Line Perpendicular to a Given Line Passing Through a Given Point

Find the equation of the line perpendicular to 5x−3y+4=05𝑥−3𝑦+4=0 and passing through the point (−4,1).(−4,1).

MEDIA

Access these online resources for additional instruction and practice with linear equations.

• Solving rational equations
• Equation of a line given two points
• Finding the equation of a line perpendicular to another line through a given point
• Finding the equation of a line parallel to another line through a given point

2.2 Section Exercises

Verbal

What does it mean when we say that two lines are parallel?

What is the relationship between the slopes of perpendicular lines (assuming neither is horizontal nor vertical)?

How do we recognize when an equation, for example y=4x+3,𝑦=4𝑥+3, will be a straight line (linear) when graphed?

What does it mean when we say that a linear equation is inconsistent?

When solving the following equation:

2x−5=4x+12𝑥−5=4𝑥+1

explain why we must exclude x=5𝑥=5 and x=−1𝑥=−1 as possible solutions from the solution set.

Algebraic

For the following exercises, solve the equation for x.𝑥.

1. 7x+2=3x−97𝑥+2=3𝑥−9
2. 4x−3=54𝑥−3=5
3. 3(x+2)−12=5(x+1)3(𝑥+2)−12=5(𝑥+1)
4. 12−5(x+3)=2x−512−5(𝑥+3)=2𝑥−5
5. 12−13x=4312−13𝑥=43
6. x3−34=2x+312𝑥3−34=2𝑥+312
7. 23x+12=31623𝑥+12=316
8. 3(2x−1)+x=5x+33(2𝑥−1)+𝑥=5𝑥+3
9. 2×3−34=x6+2142𝑥3−34=𝑥6+214
10. x+24−x−13=2𝑥+24−𝑥−13=2

For the following exercises, solve each rational equation for x.𝑥. State all x-values that are excluded from the solution set.

1. 3x−13=163𝑥−13=16
2. 2−3x+4=x+2x+42−3𝑥+4=𝑥+2𝑥+4
3. 3x−2=1x−1+7(x−1)(x−2)3𝑥−2=1𝑥−1+7(𝑥−1)(𝑥−2)
4. 3xx−1+2=3x−13𝑥𝑥−1+2=3𝑥−1
5. 5x+1+1x−3=−6×2−2x−35𝑥+1+1𝑥−3=−6𝑥2−2𝑥−3
6. 1x=15+32×1𝑥=15+32𝑥

For the following exercises, find the equation of the line using the point-slope formula. Write all the final equations using the slope-intercept form.

(0,3)(0,3) with a slope of 2323

(1,2)(1,2) with a slope of −45−45

x-intercept is 1, and (−2,6)(−2,6)

y-intercept is 2, and (4,−1)(4,−1).

(−3,10)(−3,10) and (5,−6)(5,−6)

(1,3) and  (5,5)(1,3) and  (5,5)

parallel to y=2x+5𝑦=2𝑥+5 and passes through the point (4,3)(4,3).

perpendicular to 3y=x−43𝑦=𝑥−4 and passes through the point (−2,1)(−2,1) .

For the following exercises, find the equation of the line using the given information.

(−2,0)(−2,0) and (−2,5)(−2,5)

(1,7)(1,7) and (3,7)(3,7)

The slope is undefined and it passes through the point (2,3).(2,3).

The slope equals zero and it passes through the point (1,−4).(1,−4).

The slope is 3434 and it passes through the point (1,4)(1,4).

(–1,3)(–1,3) and (4,–5)(4,–5)

Graphical

For the following exercises, graph the pair of equations on the same axes, and state whether they are parallel, perpendicular, or neither.

• y=2x+7y=−12x−4𝑦=2𝑥+7𝑦=−12𝑥−4

• 3x−2y=56y−9x=63𝑥−2𝑦=56𝑦−9𝑥=6

• y=3x+14y=3x+2𝑦=3𝑥+14𝑦=3𝑥+2

• x=4y=−3𝑥=4𝑦=−3

Numeric

For the following exercises, find the slope of the line that passes through the given points.

1. (5,4)(5,4) and (7,9)(7,9)
2. (−3,2)(−3,2) and (4,−7)(4,−7)
3. (−5,4)(−5,4) and (2,4)(2,4)
4. (−1,−2)(−1,−2) and (3,4)(3,4)
5. (3,−2)(3,−2) and (3,−2)(3,−2)
6. For the following exercises, find the slope of the lines that pass through each pair of points and determine whether the lines are parallel or perpendicular.

(−1,3) and  (5,1)(−2,3) and  (0,9)(−1,3) and  (5,1)(−2,3) and  (0,9)

(2,5) and  (5,9)(−1,−1) and  (2,3)(2,5) and  (5,9)(−1,−1) and  (2,3)

Technology

For the following exercises, express the equations in slope intercept form (rounding each number to the thousandths place). Enter this into a graphing calculator as Y1, then adjust the ymin and ymax values for your window to include where the y-intercept occurs. State your ymin and ymax values.

• 0.537x−2.19y=1000.537𝑥−2.19𝑦=100

• 4,500x−200y=9,5284,500𝑥−200𝑦=9,528

• 200−30yx=70200−30𝑦𝑥=70

Extensions

Starting with the point-slope formula y−y1=m(x−x1),𝑦−𝑦1=𝑚(𝑥−𝑥1), solve this expression for x𝑥 in terms of x1,y,y1,𝑥1,𝑦,𝑦1, and m𝑚.

Starting with the standard form of an equation Ax+By=C𝐴𝑥+𝐵𝑦=𝐶 solve this expression for y𝑦 in terms of A,B,C𝐴,𝐵,𝐶 and x𝑥. Then put the expression in slope-intercept form.

Use the above derived formula to put the following standard equation in slope intercept form: 7x−5y=25.7𝑥−5𝑦=25.

Given that the following coordinates are the vertices of a rectangle, prove that this truly is a rectangle by showing the slopes of the sides that meet are perpendicular.

(–1,1),(2,0),(3,3)(–1,1),(2,0),(3,3) and (0,4)(0,4)

Find the slopes of the diagonals in the previous exercise. Are they perpendicular?

Real-World Applications

The slope for a wheelchair ramp for a home has to be 112.112. If the vertical distance from the ground to the door bottom is 2.5 ft, find the distance the ramp has to extend from the home in order to comply with the needed slope.

1. If the profit equation for a small business selling x𝑥 number of item one and y𝑦 number of item two is p=3x+4y,𝑝=3𝑥+4𝑦, find the y𝑦 value when p=$453and  x=75.𝑝=$453and  𝑥=75.
2. For the following exercises, use this scenario: The cost of renting a car is $45/wk plus $0.25/mi traveled during that week. An equation to represent the cost would be y=45+.25x,𝑦=45+.25𝑥, where x𝑥 is the number of miles traveled.
3. What is your cost if you travel 50 mi?
4. If your cost were $63.75,$63.75, how many miles were you charged for traveling?
5. Suppose you have a maximum of $100 to spend for the car rental. What would be the maximum number of miles you could travel?