QuestionFind an equation for the line below. Continue

Studdy Solution

STEP 1

What is this asking? We need to find the equation of a line given two points on the graph. Watch out! Don't mix up the $x$ and $y$ coordinates!
Also, double-check your calculations to avoid simple arithmetic errors.

STEP 2

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

STEP 3

Remember, the slope, often denoted by $m$ , tells us how steep the line is.
It's the rise over run, or the change in $y$ divided by the change in $x$ .

STEP 4

Our two points are $(-4, -2)$ and $(4, 4)$ .
Let's label them to make things super clear!
We'll call $(-4, -2)$ point 1, so $x_1 = -4$ and $y_1 = -2$ .
And we'll call $(4, 4)$ point 2, so $x_2 = 4$ and $y_2 = 4$ .

STEP 5

The slope formula is: $m = \frac{y_2 - y_1}{x_2 - x_1}$ Let's plug in our values: $m = \frac{4 - (-2)}{4 - (-4)}$ Notice the double negatives!
Subtracting a negative is the same as adding a positive.

STEP 6

So, we have: $m = \frac{4 + 2}{4 + 4} = \frac{6}{8}$ We can simplify this fraction by dividing both the numerator and the denominator by their greatest common factor, which is 2: $m = \frac{6 \div 2}{8 \div 2} = \frac{3}{4}$ Awesome! Our slope is $\frac{3}{4}$ .
This means that for every 4 units we move to the right along the $x$ -axis, we move up 3 units along the $y$ -axis.

STEP 7

The y-intercept is the point where the line crosses the $y$ -axis.
It's the value of $y$ when $x = 0$ .
We can use the slope-intercept form of a linear equation, which is $y = mx + b$ , where $m$ is the slope and $b$ is the y-intercept.

STEP 8

We already know the slope ( $m = \frac{3}{4}$ ) and we can use one of our points to find $b$ .
Let's use point 2, $(4, 4)$ : $4 = \frac{3}{4} \cdot 4 + b$

STEP 9

Multiplying $\frac{3}{4}$ by 4 gives us 3: $4 = 3 + b$

STEP 10

To isolate $b$ , we subtract 3 from both sides of the equation: $4 - 3 = 3 + b - 3$ $1 = b$ So, our y-intercept is 1!

STEP 11

Now we have everything we need!
We know the slope ( $m = \frac{3}{4}$ ) and the y-intercept ( $b = 1$ ).
We can plug these values into the slope-intercept form, $y = mx + b$ , to get our final equation: $y = \frac{3}{4}x + 1$

STEP 12

The equation of the line is $y = \frac{3}{4}x + 1$ .

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