Math  /  Geometry

QuestionRead carefully and choose the name of the student who made the correct statement. * 1 point
CARD 2: Mrs. Hyde writes an equation and a coordinate point on the board. She asks her students to write the equation of the line that is perpendicular to the given line and passes through the given point. y=23x+8y=\frac{2}{3} x+8 (8,9)(-8,9)
Jand says the equation is y=32x+21y=-\frac{3}{2} x+21
Jayden says the equation is y=32x3y=-\frac{3}{2} x-3 Jana Jayden

Studdy Solution

STEP 1

What is this asking? Which student correctly found the equation of a line perpendicular to y=23x+8y = \frac{2}{3}x + 8 and passing through the point (8,9)(-8, 9)? Watch out! Remember, perpendicular lines have negative reciprocal slopes!
Don't forget to use the point-slope form to find the new equation.

STEP 2

1. Find the perpendicular slope.
2. Use the point-slope form.
3. Simplify to slope-intercept form.

STEP 3

The **given line** has a slope of 23\frac{2}{3}.
A line perpendicular to this will have a slope that is the negative reciprocal.

STEP 4

To find the negative reciprocal, we flip the fraction and change the sign.
So, the **perpendicular slope** is 32-\frac{3}{2}.
Flipping 23\frac{2}{3} gives us 32\frac{3}{2}, and changing the sign makes it 32-\frac{3}{2}.

STEP 5

We know the **perpendicular slope** is 32-\frac{3}{2} and the line passes through the point (8,9)(-8, 9).
We can use the **point-slope form** of a linear equation: yy1=m(xx1)y - y_1 = m(x - x_1), where (x1,y1)(x_1, y_1) is our point and mm is our slope.

STEP 6

Plugging in our values, we get y9=32(x(8))y - 9 = -\frac{3}{2}(x - (-8)).
Notice how we carefully substitute the values, remembering that subtracting a negative is the same as adding.

STEP 7

Let's simplify our equation y9=32(x+8)y - 9 = -\frac{3}{2}(x + 8).
First, distribute the 32-\frac{3}{2} to both terms inside the parentheses: y9=32x328y - 9 = -\frac{3}{2}x - \frac{3}{2} \cdot 8.

STEP 8

Simplifying further, we get y9=32x12y - 9 = -\frac{3}{2}x - 12.
We multiplied 32\frac{3}{2} by 88 to get 1212.
Since it was 32-\frac{3}{2}, the result is 12-12.

STEP 9

Now, let's add 99 to both sides of the equation to isolate yy: y9+9=32x12+9y - 9 + 9 = -\frac{3}{2}x - 12 + 9.

STEP 10

This simplifies to y=32x3y = -\frac{3}{2}x - 3.
Adding 99 to 12-12 gives us 3-3.

STEP 11

Jayden is correct!
His equation, y=32x3y = -\frac{3}{2}x - 3, matches the equation we derived.

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