4.3 Q-10

Question Statement

Find the equation of the following lines:

a) Through the point $A(-6, 5)$ with slope 7.
b) Through the point $(8, -3)$ with slope 0.
c) Through the point $(-8, 5)$ with an undefined slope.
d) Through the points $(-5, -3)$ and $(9, -1)$ .
e) Given the y-intercept of -7 and a slope of 5.
f) Given the x-intercept of -9 and a slope of 4.
g) Given the x-intercept of -3 and the y-intercept of 4.

Background and Explanation

To find the equation of a line, we use different forms depending on the given information:

Point-Slope Form: $y - y_1 = m(x - x_1)$ , where $m$ is the slope and $(x_1, y_1)$ is a point on the line.
Slope-Intercept Form: $y = mx + c$ , where $m$ is the slope and $c$ is the y-intercept.
Intercept Form: $\frac{x}{a} + \frac{y}{b} = 1$ , where $a$ and $b$ are the x- and y-intercepts, respectively.

Solution

Part (a): Through $A(-6, 5)$ with slope 7

We apply the point-slope formula:
$y - 5 = 7(x - (-6))$
Simplify the equation:
$y - 5 = 7(x + 6)$
Expand and simplify:
$y - 5 = 7x + 42$
$y = 7x + 47$
Rearrange into standard form:
$7x - y + 47 = 0$

Part (b): Through $(8, -3)$ with slope 0

Since the slope is 0, the equation represents a horizontal line. The equation of a horizontal line is simply $y = \text{constant}$ , where the constant is the y-coordinate of the point:
$y = -3$

Part (c): Through $(-8, 5)$ with undefined slope

An undefined slope means the line is vertical. The equation of a vertical line is $x = \text{constant}$ , where the constant is the x-coordinate of the point:
$x = -8$

Part (d): Through $(-5, -3)$ and $(9, -1)$

First, calculate the slope using the formula:
$m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{-1 + 3}{9 + 5} = \frac{2}{14} = \frac{1}{7}$

Now, use the point-slope form with point $(-5, -3)$ :
$y + 3 = \frac{1}{7}(x + 5)$
Multiply both sides by 7 to eliminate the fraction:
$7(y + 3) = x + 5$
Expand and simplify:
$7y + 21 = x + 5$
Rearrange into standard form:
$x - 7y - 16 = 0$

Part (e): Given y-intercept = -7, slope = 5

We use the slope-intercept form $y = mx + c$ , where $m = 5$ and $c = -7$ :
$y = 5x - 7$
Rearrange into standard form:
$5x - y - 7 = 0$

Part (f): Given x-intercept = -9, slope = 4

At the x-axis, $y = 0$ . Substitute $y = 0$ and $x = -9$ into the equation to find $c$ :
$0 = 4(-9) + c$
$0 = -36 + c$
$c = 36$

Now, use the slope-intercept form:
$y = 4x + 36$
Rearrange into standard form:
$4x + y + 36 = 0$

Part (g): Given x-intercept = -3, y-intercept = 4

Use the intercept form $\frac{x}{a} + \frac{y}{b} = 1$ , where $a = -3$ and $b = 4$ :
$\frac{x}{-3} + \frac{y}{4} = 1$
Multiply through by -12 to eliminate fractions:
$4x - 3y = -12$
Rearrange into standard form:
$4x - 3y + 12 = 0$

Key Formulas or Methods Used

Point-Slope Form:
$y - y_1 = m(x - x_1)$
Used when you know a point on the line and the slope.
Slope-Intercept Form:
$y = mx + c$
Used when you know the slope and the y-intercept.
Intercept Form:
$\frac{x}{a} + \frac{y}{b} = 1$
Used when you know the x- and y-intercepts.

Summary of Steps

For part (a), use point-slope form with the given point and slope: $7x - y + 47 = 0$ .
For part (b), use the fact that the slope is 0 for a horizontal line: $y = -3$ .
For part (c), the undefined slope indicates a vertical line: $x = -8$ .
For part (d), calculate the slope, then use point-slope form to find the equation: $x - 7y - 16 = 0$ .
For part (e), use the slope-intercept form: $5x - y - 7 = 0$ .
For part (f), find $c$ using the x-intercept, then use slope-intercept form: $4x + y + 36 = 0$ .
For part (g), use intercept form and simplify to get the equation: $4x - 3y + 12 = 0$ .