Math  /  Algebra

QuestionWrite the equation of a line in slope-intercept form with the following characteristics: A.) slope =1/4;y=1 / 4 ; y-intercept =2=2 B.) slope =3/2=-3 / 2; passes through (4,7)(-4,7) C.) passes through (2,1)(-2,1) and (2,5)(2,-5)

Studdy Solution

STEP 1

What is this asking? We need to find the equations of lines, written as y=mx+by = mx + b, given some info about them! Watch out! Remember, the *y*-intercept is where the line crosses the *y*-axis, and it's a point (0,b)(0, b).

STEP 2

1. Solve Part A
2. Solve Part B
3. Solve Part C

STEP 3

We're given the **slope** m=14m = \frac{1}{4} and the **y-intercept** b=2b = 2.
Since slope-intercept form is y=mx+by = mx + b, we can **plug in** our values directly!

STEP 4

So, the equation is y=14x+2y = \frac{1}{4}x + 2.
Boom!

STEP 5

This time, we have the **slope** m=32m = -\frac{3}{2} and a **point** (4,7)(-4, 7).
We can use the point-slope form of a line: yy1=m(xx1)y - y_1 = m(x - x_1), where (x1,y1)(x_1, y_1) is our given point.

STEP 6

**Substitute** the given values: y7=32(x(4))y - 7 = -\frac{3}{2}(x - (-4)).
Notice how the subtraction of a negative four becomes addition!

STEP 7

**Simplify**: y7=32(x+4)y - 7 = -\frac{3}{2}(x + 4).
Now **distribute** the 32-\frac{3}{2}: y7=32x6y - 7 = -\frac{3}{2}x - 6.

STEP 8

Finally, **add** 77 to both sides to **isolate** *y*: y=32x+1y = -\frac{3}{2}x + 1.
Done!

STEP 9

We have two points: (2,1)(-2, 1) and (2,5)(2, -5).
First, let's **find the slope** using the formula m=y2y1x2x1m = \frac{y_2 - y_1}{x_2 - x_1}.

STEP 10

**Substitute** the coordinates: m=512(2)m = \frac{-5 - 1}{2 - (-2)}.
This simplifies to m=64m = \frac{-6}{4}, which further reduces to m=32m = -\frac{3}{2}.

STEP 11

Now we have a **slope** and two points!
We can pick either point and use the point-slope form.
Let's use (2,1)(-2, 1).

STEP 12

So we have y1=32(x(2))y - 1 = -\frac{3}{2}(x - (-2)), which becomes y1=32(x+2)y - 1 = -\frac{3}{2}(x + 2).

STEP 13

**Distribute**: y1=32x3y - 1 = -\frac{3}{2}x - 3.

STEP 14

**Add** 11 to both sides: y=32x2y = -\frac{3}{2}x - 2.
Fantastic!

STEP 15

A) y=14x+2y = \frac{1}{4}x + 2 B) y=32x+1y = -\frac{3}{2}x + 1 C) y=32x2y = -\frac{3}{2}x - 2

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