1. Use point-slope form to write the equation of a line that has a slope of 2/3 and passes through (-3, -1). Write your final equation in slope-intercept form.
2. Write the equation in standard form using integers (no fractions or decimals): π¦ = β2/3π₯ β 1
3. Write an equation of the line that passes through (2, -1) and is parallel to the graph of y = 5x β 2. Write your final equation in slope-intercept form.
4. Write an equation of the line that passes through(3, 5) and is perpendicular to the graph of y = β3x +7. Write your final equation in slope-intercept form.
5. Sketch a graph y = |x β 3| β 2 and describe the translations.
1. Sure! So we have a slope of 2/3 and a point (-3, -1). Let's plug it into the point-slope form:
y - y1 = m(x - x1)
Let's substitute the values:
y - (-1) = 2/3(x - (-3))
Simplifying further:
y + 1 = 2/3(x + 3)
Now, let's bring it to the slope-intercept form (y = mx + b):
y + 1 = 2/3x + 2
Subtract 1 from both sides:
y = 2/3x + 1
2. Alrighty then! We have the equation y = -2/3x - 1. Let's convert it to standard form:
Multiply the equation by -3 to get rid of the fraction:
-3y = 2x + 3
Now, let's rearrange it so that the equation is in the standard form (Ax + By = C):
2x + 3 + 3y = 0
And voila! The equation in standard form is: 2x + 3 + 3y = 0.
3. Alright, we got a line parallel to y = 5x - 2 and passing through (2, -1). Let's find the slope of the given line, which is 5. Since parallel lines have the same slope, our new line will also have a slope of 5.
Now let's plug the values into the slope-intercept form (y = mx + b):
-1 = 5(2) + b
Simplify further:
-1 = 10 + b
Subtract 10 from both sides:
b = -11
So, our final equation in slope-intercept form is: y = 5x - 11.
4. Alrighty then! We have a line perpendicular to y = -3x + 7 and passing through (3, 5). The slope of the given line is -3. And since perpendicular lines have slopes that are negative reciprocals, the slope of our new line will be 1/3.
Plugging the values into the slope-intercept form (y = mx + b):
5 = 1/3(3) + b
Simplify further:
5 = 1 + b
Subtract 1 from both sides:
b = 4
So, our final equation in slope-intercept form is: y = 1/3x + 4.
5. Ah, the absolute value function is no joke! The graph of y = |x - 3| - 2 is a V-shaped graph with the vertex at (3, -2). The absolute value function takes the distance between x and 3, and then subtracts 2 from it. This translates the V-shaped graph two units downward.
So, the graph is a downward-shifted V-shape centered at (3, -2). Keep smiling!
1. To write the equation of a line using the point-slope form, we use the formula:
y - y1 = m(x - x1)
where m is the slope of the line and (x1, y1) is a point on the line. In this case, the slope is 2/3 and the point is (-3, -1). Plugging in these values into the formula, we get:
y - (-1) = (2/3)(x - (-3))
Simplifying further, we have:
y + 1 = (2/3)(x + 3)
To write the equation in slope-intercept form (y = mx + b), we need to isolate y. Let's continue simplifying:
y + 1 = (2/3)(x + 3)
Distributing (2/3) to (x + 3):
y + 1 = (2/3)x + 2
Subtracting 1 from both sides:
y = (2/3)x + 1 - 1
Simplifying further:
y = (2/3)x
Therefore, the equation of the line in slope-intercept form is y = (2/3)x.
2. To write the equation in standard form, we multiply both sides of the equation by 3 to eliminate the fraction:
3y = -2x - 3
Rearranging the equation:
2x + 3y = -3
Therefore, the equation in standard form, using integers, is 2x + 3y = -3.
3. Since the line is parallel to y = 5x - 2, it will have the same slope. We can use the point-slope form to find the equation of the line passing through (2, -1) with a slope of 5:
y - (-1) = 5(x - 2)
Simplifying further:
y + 1 = 5x - 10
Subtracting 1 from both sides:
y = 5x - 11
Therefore, the equation of the line in slope-intercept form is y = 5x - 11.
4. To find the equation of the line perpendicular to y = -3x + 7, we need to find the negative reciprocal of the slope (-3). The negative reciprocal of -3 is 1/3. Using the point-slope form with the point (3, 5):
y - 5 = (1/3)(x - 3)
Simplifying further:
y - 5 = (1/3)x - 1
Adding 5 to both sides:
y = (1/3)x + 4
Therefore, the equation of the line in slope-intercept form is y = (1/3)x + 4.
5. The graph of y = |x - 3| - 2 represents an absolute value function. The vertical translation -2 shifts the graph downward by 2 units. The graph has a V-shape and the vertex occurs at (3, -2). The slope of the lines on either side of the vertex is 1, resulting in a steep slope on the right side of the vertex and a steep slope on the left side of the vertex.
1. To write the equation of a line using point-slope form, you will need the slope of the line and the coordinates of a point that the line passes through. The point-slope form of a linear equation is given by the equation: y - y1 = m(x - x1), where (x1, y1) represents the coordinates of the given point, and m represents the slope of the line.
In this case, the slope is given as 2/3 and the point the line passes through is (-3, -1). Plugging these values into the point-slope form equation, we get: y - (-1) = (2/3)(x - (-3)).
Simplifying this equation, we have: y + 1 = (2/3)(x + 3).
To write this equation in slope-intercept form (y = mx + b), we need to isolate the y variable. Distributing 2/3 to (x + 3), we get: y + 1 = (2/3)x + 2.
Subtracting 1 from both sides, we have: y = (2/3)x + 1.
Therefore, the equation of the line in slope-intercept form is y = (2/3)x + 1.
2. To write the equation in standard form using integers, we need to eliminate the fraction. Start with the equation given: y = -2/3x - 1.
We know that any fraction can be written as a division, so we can re-write the equation as follows:
3y = -2x - 3.
To eliminate the fraction, we multiply every term in the equation by the denominator of the fraction, which, in this case, is 3.
So, the equation becomes: 3y = -2x - 3. Multiplying further, we get:
3y + 2x = -3.
To write the equation in standard form, we typically arrange the variables in a specific order and make sure the coefficients are integers. In this case, to arrange the variables in a specific order, we can write the equation as:
2x + 3y = -3.
This equation is now in standard form, using integers instead of fractions or decimals.
3. To find the equation of a line that is parallel to another line, we need to use the fact that parallel lines have the same slope. The given equation is y = 5x - 2, which means the slope of this line is 5.
Since we need to find a line parallel to this with a point on the line (2, -1), we can use the point-slope form again.
Using the point-slope form equation, we have:
y - (-1) = 5(x - 2).
Simplifying this equation, we get:
y + 1 = 5x - 10.
To write this equation in slope-intercept form (y = mx + b), we need to isolate the y variable. Subtracting 1 from both sides, we have:
y = 5x - 11.
Therefore, the equation of the line that passes through (2, -1) and is parallel to the graph y = 5x - 2 in slope-intercept form is y = 5x - 11.
4. To find the equation of a line perpendicular to another line, we need to use the fact that perpendicular lines have negative reciprocal slopes. The given equation is y = -3x + 7, which means the slope of this line is -3.
To find the negative reciprocal of -3, we take the reciprocal, which is -1/3, and change the sign. So the slope of the perpendicular line is 1/3.
Using the point-slope form equation again, we have:
y - 5 = (1/3)(x - 3).
Simplifying this equation, we get:
y - 5 = (1/3)x - 1.
To write this equation in slope-intercept form (y = mx + b), we need to isolate the y variable. Adding 5 to both sides, we have:
y = (1/3)x + 4.
Therefore, the equation of the line that passes through (3, 5) and is perpendicular to the graph y = -3x + 7 in slope-intercept form is y = (1/3)x + 4.
5. The equation y = |x - 3| - 2 describes an absolute value function. To sketch its graph, we first need to understand the impact of the absolute value and the translations involved.
The absolute value function |x| represents the distance between x and zero, so |x - 3| is the distance between x and 3. Subtracting 2 vertically shifts the graph down by 2 units.
To sketch the graph, we start by noting that the vertex, the lowest point of the "V" shape, occurs when x - 3 = 0, giving x = 3. Therefore, the vertex is at (3, -2), which represents the point (x, y) = (3, -2).
Next, we choose points on either side of the vertex to plot additional points. For example, using x = 4 and x = 2, we find y values of 1 and 1, respectively, using the equation y = |x - 3| - 2.
Finally, we plot these points and connect them smoothly. Our graph should look like a downward-opening "V" shape with the vertex at (3, -2) and the graph translated downward by 2 units compared to |x - 3|.
I am not going to do all your work for you.
I will start number 4
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4. Write an equation of the line that passes through(3, 5) and is perpendicular to the graph of y = β3x +7. Write your final equation in slope-intercept form.
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what is the slope of a line perpendicular to y = -3x + 7 ????
well m = -3
the slope of a perpendicular = -1/m = -(1/-3)= 1/3
so we have
y = (1/3) x + b
what is b ????
well we havea point (3,5)
so
5 =(1/3)(3) + b
5 = 1 + b
b= 4
so in the end
y = (1/3) x + 4