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Math  /  Algebra

QuestionGiven the line y=3x−1y=3 x-1
What is the slope of a line parallel to this line? m=\mathrm{m}= □\square
What is the equation of the line parallel to the given line that passes through the point (−1,3)(-1,3) ? y′′=y^{\prime \prime}= □\square

Studdy Solution

STEP 1

1. The equation of the given line is in the slope-intercept form y=mx+b y = mx + b .
2. Lines that are parallel have the same slope.
3. To find the equation of a line parallel to another, we use the same slope and a point it passes through.

STEP 2

1. Identify the slope of the given line.
2. Determine the slope of a line parallel to the given line.
3. Use the point-slope form to find the equation of the parallel line that passes through a given point.

STEP 3

Identify the slope of the given line y=3x−1 y = 3x - 1 .
The slope-intercept form of a line is y=mx+b y = mx + b , where m m is the slope.
For the given line, m=3 m = 3 .

STEP 4

The slope of a line parallel to the given line is the same as the slope of the given line.
Therefore, the slope m=3 \mathrm{m} = 3 .

STEP 5

To find the equation of the line parallel to the given line that passes through the point (−1,3)(-1, 3), use the point-slope form of a line:
y−y1=m(x−x1) y - y_1 = m(x - x_1)
where (x1,y1) (x_1, y_1) is the point the line passes through and m m is the slope.
Substitute m=3 m = 3 , x1=−1 x_1 = -1 , and y1=3 y_1 = 3 :
y−3=3(x+1) y - 3 = 3(x + 1)

STEP 6

Simplify the equation:
y−3=3x+3 y - 3 = 3x + 3
Add 3 to both sides to solve for y y :
y=3x+6 y = 3x + 6
Therefore, the equation of the line parallel to the given line that passes through the point (−1,3)(-1, 3) is:
y′′=3x+6 y^{\prime \prime} = 3x + 6

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