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Math  /  Geometry

QuestionFind an equation of the line passing through the points. (6,−1),(−12,11)(6,-1),(-12,11) \qquad
Sketch the line. WebAssign. Graphing Tool

Studdy Solution

STEP 1

What is this asking? We need to find the equation of a line that goes through two given points and then draw that line! Watch out! Remember that the equation of a line can be written in several forms, so make sure to double-check what form the problem wants!
Also, be careful with your signs when calculating the slope!

STEP 2

1. Calculate the slope.
2. Use point-slope form.
3. Convert to slope-intercept form.
4. Sketch the line.

STEP 3

Let's **define** our two points as (x1,y1)=(6,−1) (x_1, y_1) = (6, -1) and (x2,y2)=(−12,11) (x_2, y_2) = (-12, 11) .
We'll use these to find the **slope**, which tells us how steep our line is.

STEP 4

The **formula** for the slope, usually represented by m m , is: m=y2−y1x2−x1 m = \frac{y_2 - y_1}{x_2 - x_1} This formula tells us how much the *y*-value changes for every change in the *x*-value.
It's like figuring out how much you climb vertically for every step you take horizontally!

STEP 5

**Plugging in** our values, we get: m=11−(−1)−12−6 m = \frac{11 - (-1)}{-12 - 6} Remember those negative signs!
Subtracting a negative is the same as adding a positive.

STEP 6

**Simplifying**, we have: m=12−18 m = \frac{12}{-18} We can **simplify** this fraction by dividing both the numerator and the denominator by their **greatest common divisor**, which is 6. m=12÷6−18÷6=2−3=−23 m = \frac{12 \div 6}{-18 \div 6} = \frac{2}{-3} = -\frac{2}{3} So, our **slope** is m=−23 m = -\frac{2}{3} .
This means for every 3 units we move to the right, we go down 2 units!

STEP 7

Now that we have our **slope**, we can use the **point-slope form** of a linear equation, which is: y−y1=m(x−x1) y - y_1 = m(x - x_1) This form is super useful because we can plug in the slope and one of our points to get the equation!

STEP 8

Let's use the point (6,−1) (6, -1) and our **slope** m=−23 m = -\frac{2}{3} . **Plugging** these values into the point-slope form, we get: y−(−1)=−23(x−6) y - (-1) = -\frac{2}{3}(x - 6)

STEP 9

Let's make this equation look even nicer by converting it to **slope-intercept form**, which is y=mx+b y = mx + b , where m m is the slope and b b is the *y*-intercept.
This form makes it easy to see the slope and where the line crosses the *y*-axis.

STEP 10

Starting with our equation: y+1=−23(x−6) y + 1 = -\frac{2}{3}(x - 6) We **distribute** the −23 -\frac{2}{3} to both terms inside the parentheses: y+1=−23x+23⋅6 y + 1 = -\frac{2}{3}x + \frac{2}{3} \cdot 6 y+1=−23x+4 y + 1 = -\frac{2}{3}x + 4

STEP 11

Now, we **subtract** 1 from both sides of the equation to isolate y y : y=−23x+4−1 y = -\frac{2}{3}x + 4 - 1 y=−23x+3 y = -\frac{2}{3}x + 3 Awesome! Our equation in slope-intercept form is y=−23x+3 y = -\frac{2}{3}x + 3 .

STEP 12

To **sketch** the line, we can use the two points we already have, (6,−1) (6, -1) and (−12,11) (-12, 11) .
Plot these points on a graph.

STEP 13

Draw a straight line that passes through both points.
Since our slope is negative, the line should be going downwards from left to right.
The line should also cross the *y*-axis at y=3 y = 3 , which is our *y*-intercept!

STEP 14

The equation of the line passing through the points (6,−1) (6, -1) and (−12,11) (-12, 11) is y=−23x+3 y = -\frac{2}{3}x + 3 .
The sketch should show a line passing through the given points with a negative slope and a *y*-intercept of 3.

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