write an equation in slope intercept form that passes through a given point and is parallel to the graph of the given equation
(3,2) y=x+5 (-2,5), y =-4x+2 and (4,-6), y=-3/4x+1
I will do the 2nd, follow the same steps for the others
(-2,5), y =-4x+2
Since your new line is parallel to the given one, their slopes must be the same, that is, the equations will differ only in the y-intercept
let the new equation be
y = -4x + b
but (-2,5) lies on it, so
5 = -4(-2) + b
-3 = b
new equation: y = -4x - 3
To write an equation in slope-intercept form that passes through a given point and is parallel to the given equation, we need to use the fact that parallel lines have the same slope.
For the equation y = x + 5 which passes through the point (3,2), the slope-intercept form is y = mx + b, where m is the slope and b is the y-intercept. Here, the slope is 1, so our equation will have a slope of 1 as well.
Using the point-slope form, we have:
y - y1 = m(x - x1)
Plugging in the values (3,2) and m = 1:
y - 2 = 1(x - 3)
y - 2 = x - 3
y = x - 3 + 2
y = x - 1
Thus, the equation that passes through (3,2) and is parallel to y = x + 5 is y = x - 1.
Now let's do the same process for the other equations:
For the equation y = -4x + 2 which passes through the point (-2,5), the slope is -4. So, the equation that is parallel to it and passes through (-2,5) will also have a slope of -4.
Using the point-slope form, we have:
y - y1 = m(x - x1)
y - 5 = -4(x - (-2))
y - 5 = -4(x + 2)
y - 5 = -4x - 8
y = -4x - 8 + 5
y = -4x - 3
Thus, the equation that passes through (-2,5) and is parallel to y = -4x + 2 is y = -4x - 3.
For the equation y = -3/4x + 1 which passes through the point (4,-6), the slope is -3/4. So, the equation that is parallel to it and passes through (4,-6) will also have a slope of -3/4.
Using the point-slope form, we have:
y - y1 = m(x - x1)
y - (-6) = -3/4(x - 4)
y + 6 = -3/4(x - 4)
y + 6 = -3/4x + 3
y = -3/4x + 3 - 6
y = -3/4x - 3
Thus, the equation that passes through (4,-6) and is parallel to y = -3/4x + 1 is y = -3/4x - 3.
To find an equation in slope-intercept form that passes through a given point and is parallel to a given graph, you need to consider two properties: the slope and the given point.
The slope-intercept form of an equation is given by:
y = mx + b
where:
m = slope of the line
b = y-intercept
To find the slope of the given equation, you can observe the coefficient of x. For example, in the equation y = x + 5, the slope (m) is 1.
Since you need to find a line that is parallel to the given equation, it will have the same slope as the given equation.
Step 1: Find the slope (m) of the given equation.
Given equation: y = x + 5
Slope (m) = 1
Step 2: Use the given point and slope to write the equation.
Given point: (3,2)
Using the point-slope formula, we have:
y - y1 = m(x - x1)
Substitute the values:
y - 2 = 1(x - 3)
Simplify:
y - 2 = x - 3
Rearrange the equation to fit the slope-intercept form:
y = x - 1
So, the equation in slope-intercept form that passes through the point (3,2) and is parallel to the graph of y = x + 5 is y = x - 1.
Similarly, you can follow the same process for the other two given equations to find the equations that pass through the given points and are parallel to each corresponding graph.
For (−2,5), y = −4x + 2:
Slope (m) = -4
Using the point (−2,5) and the slope -4, the equation in slope-intercept form will be:
y - 5 = -4(x - (-2))
y - 5 = -4(x + 2)
y - 5 = -4x - 8
y = -4x - 3
For (4,−6), y = −3/4x + 1:
Slope (m) = -3/4
Using the point (4,−6) and the slope -3/4, the equation in slope-intercept form will be:
y - (-6) = -3/4(x - 4)
y + 6 = -3/4(x - 4)
y + 6 = -3/4x + 3
y = -3/4x - 3
Hence, the equations in slope-intercept form that pass through the given points and are parallel to the respective given equations are:
1) For (3,2), y = x - 1
2) For (−2,5), y = -4x - 3
3) For (4,−6), y = -3/4x - 3