Math  /  Algebra

QuestionGiven the coordinates of two points (105,120) and (102,100), find the equation of the line in slope-intercept form.\text{Given the coordinates of two points } (105, 120) \text{ and } (102, 100), \text{ find the equation of the line in slope-intercept form.} Use the formula for the slope m=y2y1x2x1 to find the slope of the line.\text{Use the formula for the slope } m = \frac{y_2 - y_1}{x_2 - x_1} \text{ to find the slope of the line.} Then, use one of the points to solve for the y-intercept b by substituting the values into the equation y=mx+b.\text{Then, use one of the points to solve for the y-intercept } b \text{ by substituting the values into the equation } y = mx + b.

Studdy Solution

STEP 1

What is this asking? We need to find the equation of a line that goes through two specific points, and we want that equation to be in slope-intercept form, which looks like y=mx+by = mx + b. Watch out! Don't mix up the xx and yy coordinates when calculating the slope!
Also, remember that the slope-intercept form is y=mx+by = mx + b, not y=bx+my = bx + m.

STEP 2

1. Calculate the Slope
2. Find the Y-Intercept
3. Write the Equation

STEP 3

Let's **label our points** to avoid confusion!
We'll call (105,120)(105, 120) point 1, so x1=105x_1 = 105 and y1=120y_1 = 120.
And (102,100)(102, 100) will be point 2, making x2=102x_2 = 102 and y2=100y_2 = 100.

STEP 4

Now, we'll **plug these values** into our super-powered slope formula: m=y2y1x2x1=100120102105m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{100 - 120}{102 - 105}

STEP 5

Time to **crunch the numbers**: m=203m = \frac{-20}{-3}

STEP 6

Dividing a negative by a negative gives us a positive, awesome! m=203m = \frac{20}{3} So, our **slope** is m=203m = \frac{20}{3}.

STEP 7

We've got our slope, now we need the **y-intercept**, which we call bb.
We can use the slope-intercept form, y=mx+by = mx + b, and plug in the coordinates of one of our points.
Let's use point 1: (105,120)(105, 120), where x=105x = 105 and y=120y = 120.
We also know m=203m = \frac{20}{3}.

STEP 8

**Substitute** those values into our equation: 120=203105+b120 = \frac{20}{3} \cdot 105 + b

STEP 9

Let's **simplify**: 120=201053+b120 = \frac{20 \cdot 105}{3} + b 120=21003+b120 = \frac{2100}{3} + b120=700+b120 = 700 + b

STEP 10

To **isolate** bb, we'll subtract 700 from both sides of the equation: 120700=700700+b120 - 700 = 700 - 700 + b 580=b-580 = bSo, our **y-intercept** is b=580b = -580.

STEP 11

We have our **slope**, m=203m = \frac{20}{3}, and our **y-intercept**, b=580b = -580.
Now, we just plug them into the slope-intercept form, y=mx+by = mx + b.

STEP 12

**Substituting** the values, we get: y=203x580y = \frac{20}{3}x - 580

STEP 13

The equation of the line in slope-intercept form is y=203x580y = \frac{20}{3}x - 580.

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