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Math  /  Algebra

QuestionA system of linear equations is graphed on the xyx y-plane below.
Find the equations of the lines. Use exact numbers. y=□x−2y=\square x-2 y=0.25x+y=0.25 x+ □\square

Studdy Solution

STEP 1

1. The problem requires finding the equations of two lines on the xyxy-plane.
2. The lines are given in slope-intercept form, y=mx+by = mx + b.
3. We need to find the slope mm and the y-intercept bb for each line.

STEP 2

1. Determine the equation of the first line using the points (−8,−6)(-8, -6) and (8,−2)(8, -2).
2. Determine the equation of the second line using the points (5,4)(5, 4) and (10,−6)(10, -6).

STEP 3

Calculate the slope mm of the first line using the points (−8,−6)(-8, -6) and (8,−2)(8, -2).
m=y2−y1x2−x1=−2−(−6)8−(−8)=416=14 m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{-2 - (-6)}{8 - (-8)} = \frac{4}{16} = \frac{1}{4}

STEP 4

Use the slope m=14m = \frac{1}{4} and one of the points, say (8,−2)(8, -2), to find the y-intercept bb of the first line.
y=mx+b y = mx + b −2=14(8)+b -2 = \frac{1}{4}(8) + b −2=2+b -2 = 2 + b b=−4 b = -4
The equation of the first line is:
y=14x−4 y = \frac{1}{4}x - 4

STEP 5

Calculate the slope mm of the second line using the points (5,4)(5, 4) and (10,−6)(10, -6).
m=y2−y1x2−x1=−6−410−5=−105=−2 m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{-6 - 4}{10 - 5} = \frac{-10}{5} = -2

STEP 6

Use the slope m=−2m = -2 and one of the points, say (5,4)(5, 4), to find the y-intercept bb of the second line.
y=mx+b y = mx + b 4=−2(5)+b 4 = -2(5) + b 4=−10+b 4 = -10 + b b=14 b = 14
The equation of the second line is:
y=−2x+14 y = -2x + 14
The equations of the lines are:
1. y=14x−4 y = \frac{1}{4}x - 4
2. y=−2x+14 y = -2x + 14

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