Math  /  Algebra

Question4. A line has a slope of 56\frac{5}{6} and passes through the point (12,14)(-12,-14). a) Write the equation y=mx+by=m x+b with the given slope substituted for mm.

Studdy Solution

STEP 1

What is this asking? We're given a **line's slope** and a **point** it goes through, and we need to find its equation in *slope-intercept form*. Watch out! Don't mix up the *x* and *y* coordinates of the point!

STEP 2

1. Plug in the Slope
2. Find the y-intercept
3. Write the Full Equation

STEP 3

Alright, we're given the **slope**, which is how steep our line is!
The slope is 56\frac{5}{6}.
Remember, slope-intercept form is y=mx+by = m \cdot x + b, where mm is the **slope**.
Let's **plug in** that 56\frac{5}{6} for mm!

STEP 4

So, our equation becomes y=56x+by = \frac{5}{6} \cdot x + b.
We're getting closer to the finish line!

STEP 5

Now, we need to find bb, which is the **y-intercept**.
That's where the line crosses the y-axis.
We know the line goes through the point (12,14)(-12, -14).
This means when x=12x = -12, y=14y = -14.
Let's **substitute** these values into our equation!

STEP 6

We get 14=56(12)+b-14 = \frac{5}{6} \cdot (-12) + b.
Now we just need to solve for bb!

STEP 7

First, let's simplify the right side: 56(12)=5(12)6=606=10\frac{5}{6} \cdot (-12) = \frac{5 \cdot (-12)}{6} = \frac{-60}{6} = -10.
So, our equation is now 14=10+b-14 = -10 + b.

STEP 8

To **isolate** bb, we need to add 1010 to both sides of the equation.
This gives us 14+10=10+10+b-14 + 10 = -10 + 10 + b, which simplifies to 4=b-4 = b.
Boom! We found our **y-intercept**: b=4b = -4!

STEP 9

We've got our **slope** (m=56m = \frac{5}{6}) and our **y-intercept** (b=4b = -4).
Let's put them together in the slope-intercept form: y=mx+by = m \cdot x + b.

STEP 10

Substituting our values, we get y=56x+(4)y = \frac{5}{6} \cdot x + (-4), which simplifies to y=56x4y = \frac{5}{6} \cdot x - 4.
And there you have it!

STEP 11

The equation of the line in slope-intercept form is y=56x4y = \frac{5}{6} \cdot x - 4.

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