QuestionFind the equation of the line through the points $(3,-2)$ and $(-4,12)$ . (Hint: $y=mx+b$ )

Studdy Solution

STEP 1

1. We are given two points: $(3, -2)$ and $(-4, 12)$ .
2. We need to find the equation of the line in the slope-intercept form $y = mx + b$ .
3. The slope-intercept form requires us to find the slope $m$ and the y-intercept $b$ .

STEP 2

1. Calculate the slope $m$ of the line using the two points.
2. Use the slope and one of the points to solve for the y-intercept $b$ .
3. Write the equation of the line using the slope $m$ and y-intercept $b$ .

STEP 3

Calculate the slope $m$ using the formula:
$m = \frac{y_2 - y_1}{x_2 - x_1}$
Substitute the given points $(x_1, y_1) = (3, -2)$ and $(x_2, y_2) = (-4, 12)$ :
$m = \frac{12 - (-2)}{-4 - 3}$

STEP 4

Simplify the expression:
$m = \frac{12 + 2}{-7} = \frac{14}{-7} = -2$

STEP 5

Use the slope $m = -2$ and one of the points, say $(3, -2)$ , to find the y-intercept $b$ using the equation $y = mx + b$ .
Substitute $m = -2$ , $x = 3$ , and $y = -2$ into the equation:
$-2 = -2(3) + b$

STEP 6

Solve for $b$ :
$-2 = -6 + b$
Add 6 to both sides:
$b = 4$

STEP 7

Write the equation of the line using the slope $m = -2$ and y-intercept $b = 4$ :
$y = -2x + 4$
This is the equation of the line through the points $(3, -2)$ and $(-4, 12)$ .

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