Write the slope-intercept form of an equation of the line that passes through the given point and is perpendicular to the graph of each equation.
1.(-2,-2),y=-1/3x+9
2.(-4,-3),4x+y=7
3.(-4,1),4x+7y=6
4.(-6,-5),4x+3y=-6
These are the answer I came up with, could you review them? Thanks
1.y=3/1x+4
2.y=1/4x-2
3.y=7/4x+8
4.y=3/4x-1/2
all correct
Let's check your answers step-by-step:
1. To find the slope of the line perpendicular to the given line, we need to take the negative reciprocal of the original slope. The given equation is y = -1/3x + 9, so the original slope is -1/3. Taking the negative reciprocal, we get 3/1, which is correctly stated in your answer.
The correct equation is y = 3x + b, where b is the y-intercept. To find b, we can substitute the coordinates of the given point (-2,-2) into the equation and solve for b:
-2 = 3(-2) + b
-2 = -6 + b
b = 4
So, the correct equation for this line is y = 3x + 4. Your answer, y = 3/1x + 4, is written in slope-intercept form, but it is not simplified.
2. The given equation is 4x + y = 7. To find the slope of the line perpendicular to this equation, we need to rearrange the equation to the slope-intercept form (y = mx + b):
y = -4x + 7
The slope of the original line is -4, so the negative reciprocal is 1/4. Your answer, y = 1/4x - 2, matches this equation correctly.
3. The given equation is 4x + 7y = 6. Rearranging the equation to slope-intercept form, we get:
7y = -4x + 6
y = (-4/7)x + (6/7)
The slope of the original line is -4/7. Taking the negative reciprocal, we get -7/4. Your answer, y = 7/4x + 8, is incorrect. It seems you have made a mistake in calculating the slope and y-intercept.
4. The given equation is 4x + 3y = -6. Rearranging the equation to slope-intercept form, we get:
3y = -4x - 6
y = (-4/3)x - 2
The slope of the original line is -4/3. Taking the negative reciprocal, we get 3/4. Your answer, y = 3/4x - 1/2, is correct.
So, to summarize:
1. The correct equation is y = 3x + 4.
2. Your answer, y = 1/4x - 2, is correct.
3. The correct equation is y = -7/4x + 6/7.
4. Your answer, y = 3/4x - 1/2, is correct.
Great job on most of the answers! Just be careful with the calculations in question 3.
To find the equation of a line that is perpendicular to a given line, you need to follow a few steps:
Step 1: Find the slope of the given line.
Step 2: Take the negative reciprocal of the slope found in Step 1.
Step 3: Use the point given to find the y-intercept.
Step 4: Write the equation in slope-intercept form (y = mx + b), where m is the slope and b is the y-intercept.
Now let's go through each question and check your answer:
1. Given equation: y = -1/3x + 9
- The slope of the given line is -1/3.
- The negative reciprocal of -1/3 is 3.
- Using the point (-2, -2) to find the y-intercept:
-2 = 3*(-2) + b
-2 = -6 + b
b = 4
- Therefore, the equation of the line that passes through (-2, -2) and is perpendicular to y = -1/3x + 9 is y = 3x + 4. ✔
2. Given equation: 4x + y = 7
- Rearrange the equation in slope-intercept form (y = mx + b):
y = -4x + 7
- The slope of the given line is -4.
- The negative reciprocal of -4 is 1/4.
- Using the point (-4, -3) to find the y-intercept:
-3 = 1/4*(-4) + b
-3 = -1 + b
b = -2
- Therefore, the equation of the line that passes through (-4, -3) and is perpendicular to 4x + y = 7 is y = 1/4x - 2. ✔
3. Given equation: 4x + 7y = 6
- Rearrange the equation in slope-intercept form (y = mx + b):
y = -4/7x + 6/7
- The slope of the given line is -4/7.
- The negative reciprocal of -4/7 is 7/4.
- Using the point (-4, 1) to find the y-intercept:
1 = 7/4*(-4) + b
1 = -7 + b
b = 8
- Therefore, the equation of the line that passes through (-4, 1) and is perpendicular to 4x + 7y = 6 is y = 7/4x + 8. ✔
4. Given equation: 4x + 3y = -6
- Rearrange the equation in slope-intercept form (y = mx + b):
y = -4/3x - 2
- The slope of the given line is -4/3.
- The negative reciprocal of -4/3 is 3/4.
- Using the point (-6, -5) to find the y-intercept:
-5 = 3/4*(-6) + b
-5 = -9/2 + b
b = 1/2
- Therefore, the equation of the line that passes through (-6, -5) and is perpendicular to 4x + 3y = -6 is y = 3/4x - 1/2. ✔
Good job! Your answers are correct for all four equations.