Can you please help me with this problem:
The x-intercept of a line has a value of -3/2 and a y intercept of 3. Another line has a slope that is three times the slope of the first line and has a y-intercept that is 28 more than the first line's y-intercept. What is the solution to this system of equation?
I am very confused...please help
x-intercept -3/2 (3/2 , 0)
for 1st line:
two points are (-3/2, 0) and (0,3)
slope = (3-0)/(0+3/2)
= 3/(3/2) = 2
so equation of 1st line is y = 2x + 3
2nd line:
slope = 3(2) = 6
y intercept = 3 + 28 = 31
equation of 2nd line: y = 6x + 31
intersection:
6x + 31 = 2x+3
4x = -28
x = -7
then y = 2(-7) + 3 = -11
Certainly! Let's solve the system of equations step-by-step.
First, let's find the equation of the first line using the given information. We know that the x-intercept is at (3/2, 0) and the y-intercept is at (0, 3).
To find the equation of a line, we can use the slope-intercept form, which is y = mx + b, where m is the slope and b is the y-intercept.
The slope (m) can be found using the formula:
m = (y2 - y1) / (x2 - x1)
Using the points (3/2, 0) and (0, 3), we can calculate the slope:
m = (0 - 3) / (3/2 - 0) = -3 / (3/2) = -2
Now that we know the slope is -2 and the y-intercept is 3, we can write the equation of the first line:
y = -2x + 3
Next, let's find the equation of the second line. We are given that its slope is three times the slope of the first line (-2). So the slope of the second line is -2 * 3 = -6.
We are also given that the second line's y-intercept is 28 more than the first line's y-intercept. Since the first line's y-intercept is 3, the second line's y-intercept is 3 + 28 = 31.
Using the slope-intercept form, we can write the equation of the second line:
y = -6x + 31
Now we have two linear equations:
1) y = -2x + 3
2) y = -6x + 31
To find the solution to this system of equations, we need to find the point where these two lines intersect, which is the x-coordinate and y-coordinate that satisfy both equations.
To solve for x, we can set the right sides of the equations equal to each other:
-2x + 3 = -6x + 31
Now, let's solve for x:
-2x + 6x = 31 - 3
4x = 28
x = 7
Now that we have the x-coordinate, we can substitute it back into either of the original equations to solve for y.
Using the first equation, y = -2x + 3:
y = -2(7) + 3
y = -14 + 3
y = -11
So the solution to the system of equations is (7, -11).