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: $m=-2$.

Step 2:

Replace m in the equation $y=mx+b$ with the value given.

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

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,

$5=-2(3)+b$

$b=5+6=11$

Step 4:

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

Therefore, the equation of the line is $y=-2x+11$.

Solution

Step 1:

Calculate the slope.

${\displaystyle m=\frac{Changeinyvalue}{Changeinxvalue}=\frac{y_2-y_1}{x_2-x_1}=\frac{5-2}{4-3}=\frac{3}{1}=3}$

Step 2:

Replace m in the equation $y=mx+b$ with the calculated slope.

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

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,

$2=3(3)+b$

$b=2-9=-7$

Step 4:

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

Therefore, the equation of the line is $y=3x-7$.

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,

$y=3x-7$

$5=3(4)-7$

$=12-7$

$=5$ (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   ${\displaystyle m=\frac{y_2-y_1}{x_2-x_1}=\frac{10-6}{4-1}=\frac{4}{3}}$.

Step 2:

Let the equation of the line be   $y=mx+b$

Substituting   ${\displaystyle m=\frac{4}{3}}$,

Therefore,   ${\displaystyle y=\frac{4}{3}x+b}$

Step 3:

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

${\displaystyle y=\frac{4}{3}x+b}$

${\displaystyle 6=\frac{4}{3}(1)+b}$

${\displaystyle b=6-\frac{4}{3}=\frac{18-4}{3}=\frac{14}{3}}$

Step 4:

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

${\displaystyle y=\frac{4}{3}x+\frac{14}{3}}$   Multiplying each term by 3 and simplifying,

$3y=4x+14$   Rearranging, ensuring that the coefficient of the x term remains positive,

$4x-3y=-14$

Therefore, the equation of the line in standard form is: $4x-3y=-14$.

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 ${\displaystyle \frac{2}{3}}$ and pass through the origin.
8. Have a slope of ${\displaystyle -\frac{4}{5}}$ 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.