Math  /  Algebra

QuestionIn a certain urban area, the relationship between the number of students, xx, in thousands, and the number of schools, yy, has an xx-intercept of -4 and a slope of 34\frac{3}{4}. Write an equation for this relationship in standard form. \square xx- \square y=y= \square

Studdy Solution

STEP 1

What is this asking? We need to find the equation of a line, describing the relationship between students and schools, and write it in a specific format. Watch out! Remember standard form is Ax+By=CAx + By = C, where AA, BB, and CC are integers, and AA is positive!

STEP 2

1. Find the slope-intercept form.
2. Convert to standard form.

STEP 3

Alright, let's **start** with the slope-intercept form, which is y=mx+by = mx + b.
Remember, mm is the **slope**, and bb is the **y-intercept**!

STEP 4

We're given the **slope**, which is m=34m = \frac{3}{4}.
Awesome! So, we have y=34x+by = \frac{3}{4}x + b.

STEP 5

Now, we need the **y-intercept**, bb.
We're given the **x-intercept**, which is 4-4.
The x-intercept happens when y=0y = 0.
Let's plug that in: 0=34(4)+b0 = \frac{3}{4}(-4) + b.

STEP 6

Multiplying 34\frac{3}{4} by 4-4 gives us 3-3.
So, 0=3+b0 = -3 + b.

STEP 7

To **isolate** bb, we **add** 33 to both sides: 3=b3 = b.
So, our **y-intercept** is 33!

STEP 8

Putting it all together, our slope-intercept equation is y=34x+3y = \frac{3}{4}x + 3.
Fantastic!

STEP 9

We want to get our equation to the form Ax+By=CAx + By = C.
Let's **start** with our slope-intercept form: y=34x+3y = \frac{3}{4}x + 3.

STEP 10

First, let's **move** that 34x\frac{3}{4}x to the other side by **subtracting** 34x\frac{3}{4}x from both sides: 34x+y=3-\frac{3}{4}x + y = 3.

STEP 11

We need integer coefficients, and AA needs to be positive.
Let's **multiply** both sides of the equation by 4-4 to achieve this: (4)(34x+y)=(4)(3)(-4)(-\frac{3}{4}x + y) = (-4)(3).

STEP 12

Distributing the 4-4 gives us 3x4y=123x - 4y = -12.
Boom! Standard form achieved!

STEP 13

Our equation in standard form is 3x4y=123x - 4y = -12.

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