QuestionIn Exercises 3-4, write an equation in standard form of the line that passes through the given points. 3. 4.

Studdy Solution

STEP 1

What is this asking? We need to find the equation of the line that goes through the points $(-2, -5)$ and $(4, 4)$ , and write it in standard form. Watch out! Standard form is $Ax + By = C$ , where $A$ , $B$ , and $C$ are integers, and $A$ is positive!
Don't mix up slope-intercept form and standard form.

STEP 2

1. Find the slope.
2. Find the equation in point-slope form.
3. Convert to standard form.

STEP 3

Let's dive in and find the slope!
We have two points, $(-2, -5)$ and $(4, 4)$ .
Remember, slope is the change in $y$ divided by the change in $x$ .

STEP 4

So, our change in $y$ is $4 - (-5) = 4 + 5 = 9$ .
Our change in $x$ is $4 - (-2) = 4 + 2 = 6$ .

STEP 5

Therefore, our slope is $\frac{9}{6}$ , which simplifies to $\frac{3}{2}$ !
We divide both the numerator and the denominator by their greatest common divisor, which is 3, to get the simplified fraction.

STEP 6

Now, let's use the point-slope form, which is $y - y_1 = m(x - x_1)$ , where $m$ is the slope and $(x_1, y_1)$ is a point on the line.
We can use either of our points, but let's use $(4, 4)$ because it has no negatives!

STEP 7

Plugging in our slope of $\frac{3}{2}$ and our point $(4, 4)$ , we get $y - 4 = \frac{3}{2}(x - 4)$ .

STEP 8

Time to convert to standard form!
First, let's get rid of the fraction by multiplying both sides of the equation by 2: $2(y - 4) = 2 \cdot \frac{3}{2}(x - 4)$ .
This simplifies to $2y - 8 = 3(x - 4)$ .

STEP 9

Distribute the 3 on the right side: $2y - 8 = 3x - 12$ .

STEP 10

Now, let's move the $x$ term to the left side by subtracting $3x$ from both sides: $2y - 3x - 8 = -12$ .

STEP 11

Add 8 to both sides to move the constant term to the right: $2y - 3x = -12 + 8$ , which simplifies to $2y - 3x = -4$ .

STEP 12

Finally, to make the coefficient of $x$ positive, multiply the entire equation by $-1$ : $-1(2y - 3x) = -1(-4)$ .
This gives us our final answer in standard form: $3x - 2y = 4$ .

STEP 13

The equation of the line in standard form is $3x - 2y = 4$ .

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