8.3 Finding the Equation of a Line

In order to determine the equation of a line, we need both the slope and the y-intercept. If we do not know this information, we can determine it by following these steps:

Step 1:

If the slope is unknown, determine it by computing the rise and run between two points on the graph.

Step 2:

Replace the unknown m in the slope-intercept form of the equation with the slope value from Step 1.

Step 3:

If the y-intercept is unknown, determine it by substituting the coordinates of a point on the line for the x and y values in the slope-intercept form of the equation from Step 2, and solve for b.

Step 4:

Replace the unknown b in the slope-intercept form of the equation from Step 2 with the y-intercept value from Step 3 to arrive at the final equation of the line.

Solution

Step 1:

In this case, the slope is already known: [latex]m = -2[/latex].

Step 2:

Replace m in the equation [latex]y = mx + b[/latex] with the value given.

Substituting for m in the slope-intercept equation [latex]y = mx + b[/latex], we obtain [latex]y = -2x + b[/latex].

Step 3:

Substitute the coordinates of the given point into the equation to solve for b.

Substituting the x– and y-coordinates of the point (3, 5) into the above equation, we obtain,

[latex]5 = -2(3) + b[/latex]

[latex]b = 5 + 6 = 11[/latex]

Step 4:

Write the equation in slope-intercept form, [latex]y = mx + b[/latex], by substituting for the values of m and b determined in the steps above.

Therefore, the equation of the line is [latex]y = -2x + 11[/latex].

Solution

Step 1:

Calculate the slope.

[latex]\displaystyle{m = \frac{Change~in~y~value}{Change~in~x~value} = \frac{y_2 - y_1}{x_2 - x_1} = \frac{5 - 2}{4 - 3} = \frac{3}{1} = 3}[/latex]

Step 2:

Replace m in the equation [latex]y = mx + b[/latex] with the calculated slope.

Substituting for m in the slope-intercept equation [latex]y = mx + b[/latex], we obtain [latex]y = 3x + b[/latex].

Step 3:

Substitute the coordinates of one of the given points into the equation to solve for b.

Substituting the x– and y-coordinates of the point (3, 2) into the above equation, we obtain,

[latex]2 = 3(3) + b[/latex]

[latex]b = 2 - 9 = -7[/latex]

Step 4:

Write the equation in slope-intercept form, [latex]y = mx + b[/latex], by substituting for the values of m and b determined in the steps above.

Therefore, the equation of the line is [latex]y = 3x - 7[/latex].

Note: A good check to validate the equation is substituting the coordinates of the other point into the equation to ensure that it is a solution.
That is, substituting the x- and y-coordinates of the point (4, 5) into the equation,

[latex]y = 3x - 7[/latex]

[latex]5 = 3(4) - 7[/latex]

[latex]= 12 - 7[/latex]

[latex]= 5[/latex] (True)

Solution

Step 1:

Start by choosing any two points (with integer coordinates) on the line: e.g., (1, 6) and (4, 10).

The slope of the line is   [latex]\displaystyle{m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{10 - 6}{4 - 1} = \frac{4}{3}}[/latex].

Step 2:

Let the equation of the line be   [latex]y = mx + b[/latex]

Substituting   [latex]\displaystyle{m = \frac{4}{3}}[/latex],

Therefore,   [latex]\displaystyle{y = \frac{4}{3}x + b}[/latex]

Step 3:

Substituting the coordinates of one of the points (1, 6) into the above equation and solving for b,

[latex]\displaystyle{y = \frac{4}{3}x + b}[/latex]

[latex]\displaystyle{6 = \frac{4}{3}(1) + b}[/latex]

[latex]\displaystyle{b = 6 - \frac{4}{3} = \frac{18 - 4}{3} = \frac{14}{3}}[/latex]

Step 4:

Therefore, the equation of the line in slope-intercept form is: [latex]\displaystyle{y = \frac{4}{3}x + \frac{14}{3}}[/latex]

[latex]\displaystyle{y = \frac{4}{3}x + \frac{14}{3}}[/latex]   Multiplying each term by 3 and simplifying,

[latex]3y = 4x + 14[/latex]   Rearranging, ensuring that the coefficient of the x term remains positive,

[latex]4x - 3y = -14[/latex]

Therefore, the equation of the line in standard form is: [latex]4x - 3y = -14[/latex].

8.3 Exercises

Answers to the odd-numbered questions are available at the end of the book.

For problems 1 to 4, determine the equations of the lines in slope-intercept form that pass through the points.

1. (1, 2) and (5, 2)
2. (5, 0) and (4, 5)
3. (−3, −5) and (3, 1)
4. (−4, −7) and (5, 2)

For problems 5 to 14, determine the equations of the lines in slope-intercept form that:

5. Have a slope of −2 and pass through (2, 6).
6. Have a slope of 3 and pass through (−3, −2).
7. Have a slope of [latex]\displaystyle{\frac{2}{3}}[/latex] and pass through the origin.
8. Have a slope of [latex]\displaystyle{-\frac{4}{5}}[/latex] and pass through the origin.
9. Have an x-intercept = 4 and a y-intercept = 6.
10. Have an x-intercept = −4 and a y-intercept = 2.
11. The slope of a line is 3. The line passes through A(4, y) and B(6, 8). Find y.
12. The slope of a line is 2. The line passes through A(x, 8) and B(2, 4). Find x.
13. Points A(2, 3), B (6, 5), and C(10, y) are on a line. Find y.
14. Points D(3, 2), E (6, 5), and F(x, 1) are on a line. Find x.

For problems 15 to 18, determine the equation of the line (in standard form) for the graphs.