Math  /  Algebra

QuestionThe graph of a linear function passes through the points (1,2)(1,2) and (2,1)(2,-1).
Which equation represents the function?
A y=3x+5y=3 x+5
B y=3x5y=-3 x-5 C y=3x5y=3 x-5 D y=3x+5y=-3 x+5

Studdy Solution

STEP 1

What is this asking? We need to find the equation of a line that goes through two given points. Watch out! Don't mix up the xx and yy coordinates!
Also, remember that the slope can be positive or negative.

STEP 2

1. Find the slope.
2. Find the y-intercept.
3. Write the equation.

STEP 3

Let's **label** our points!
We'll call (1,2)(1,2) point 1, so x1=1x_1 = 1 and y1=2y_1 = 2.
And (2,1)(2,-1) will be point 2, so x2=2x_2 = 2 and y2=1y_2 = -1.

STEP 4

The **slope** is the change in yy divided by the change in xx.
The formula is: m=y2y1x2x1 m = \frac{y_2 - y_1}{x_2 - x_1} Let's plug in our **values**: m=1221 m = \frac{-1 - 2}{2 - 1} m=31 m = \frac{-3}{1} m=3 m = -3 So, our **slope** is 3-3!

STEP 5

We know the equation of a line is y=mx+by = mx + b, where mm is the **slope** and bb is the **y-intercept**.
We already found that m=3m = -3, so we have y=3x+by = -3x + b.

STEP 6

Now, let's use one of our points to find bb.
Let's use point 1: (1,2)(1,2).
We'll **substitute** x=1x = 1 and y=2y = 2 into our equation: 2=31+b 2 = -3 \cdot 1 + b 2=3+b 2 = -3 + b To **isolate** bb, we'll add 3 to both sides: 2+3=3+3+b 2 + 3 = -3 + 3 + b 5=b 5 = b So, our **y-intercept** is 55!

STEP 7

We found that the **slope** m=3m = -3 and the **y-intercept** b=5b = 5.
Putting it all together, our equation is: y=3x+5 y = -3x + 5

STEP 8

The equation that represents the function is y=3x+5y = -3x + 5, which is answer choice **D**.

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