Math  /  Algebra

QuestionA line passes through the point (10,1)(10,-1) and has a slope of 32-\frac{3}{2} Write an equation in slope-intercept form for this line. \square

Studdy Solution

STEP 1

What is this asking? We need to find the equation of a line, specifically in slope-intercept form (y=mx+by = mx + b), given a point it passes through and its slope. Watch out! Don't mix up the *x* and *y* coordinates of the given point!
Also, remember that slope-intercept form is y=mx+by = mx + b, not y=bx+my = bx + m!

STEP 2

1. Plug in the known values
2. Solve for the y-intercept
3. Write the final equation

STEP 3

We know the **slope-intercept form** of a linear equation is y=mx+by = mx + b, where mm is the **slope** and bb is the **y-intercept**.

STEP 4

We're given that the slope is 32-\frac{3}{2}, so we can **plug** that in for mm: y=32x+by = -\frac{3}{2}x + b.

STEP 5

We also know the line passes through the point (10,1)(10, -1).
This means when x=10x = 10, y=1y = -1.
Let's **substitute** these values into our equation: 1=3210+b-1 = -\frac{3}{2} \cdot 10 + b.
Now we have an equation with only one unknown, bb!

STEP 6

Let's **simplify** that equation: 1=32101+b-1 = -\frac{3}{2} \cdot \frac{10}{1} + b.
Multiplying the fractions, we get 1=302+b-1 = -\frac{30}{2} + b, which simplifies to 1=15+b-1 = -15 + b.

STEP 7

To **isolate** bb, we can add 15 to both sides of the equation: 1+15=15+b+15-1 + 15 = -15 + b + 15.
This simplifies to 14=b14 = b.
So, our **y-intercept** is 1414!

STEP 8

Now we know both the **slope**, 32-\frac{3}{2}, and the **y-intercept**, 1414.
We can **plug** these back into the slope-intercept form, y=mx+by = mx + b, to get our **final equation**: y=32x+14y = -\frac{3}{2}x + 14.

STEP 9

The equation of the line in slope-intercept form is y=32x+14y = -\frac{3}{2}x + 14.

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