Review the graphs of a system of two linear equations in two variables: x−5y=7 and 3x−2y=−4. Find the solution to both equations.
The intersection point is (__).
To find the solution to the system of equations, we can graph them and see where they intersect. Here are the steps to find the solution:
1. Write the equations in slope-intercept form (y = mx + b).
Equation 1: x - 5y = 7
Rearrange the equation to isolate y:
-5y = -x + 7
Divide by -5 on both sides:
y = (1/5)x - 7/5
Equation 2: 3x - 2y = -4
Rearrange the equation to isolate y:
-2y = -3x - 4
Divide by -2 on both sides:
y = (3/2)x + 2
2. Graph the equations using the slope-intercept form.
For Equation 1, the y-intercept is -7/5, and the slope is 1/5.
For Equation 2, the y-intercept is 2, and the slope is 3/2.
3. Plot the y-intercept for each equation on the graph.
4. Using the slope, locate the next point on each line. Draw a straight line through the two points to extend the graph.
5. The solution to the system of equations is the coordinates of the intersection point of the two lines.
After graphing the two equations, you will see that the lines intersect at a particular point. The coordinates of this intersection point are the solution to the system of equations.
Please refer to the graph or use any graphing tool to find the exact intersection point.
To find the solution to the system of equations, we can solve it using either the substitution method or the elimination method.
Let's use the elimination method:
First, we need to eliminate one variable by manipulating the equations.
We can do this by multiplying the first equation by 3 and the second equation by 1:
3(x - 5y) = 3(7)
1(3x - 2y) = 1(-4)
This simplifies to:
3x - 15y = 21
3x - 2y = -4
Now, let's subtract the second equation from the first equation:
(3x - 15y) - (3x - 2y) = 21 - (-4)
-13y = 25
Dividing both sides by -13:
y = -25/13
Now, we can substitute the value of y back into one of the original equations to solve for x:
Using the first equation, we have:
x - 5(-25/13) = 7
x + 125/13 = 7
x = 7 - 125/13
x = (91 - 125)/13
x = -34/13
Therefore, the solution to the system of equations is x = -34/13 and y = -25/13.
The intersection point of the two graphs is (-34/13, -25/13).
To find the solution to the system of linear equations, let's review their graphs first.
The first equation is x-5y=7. To graph this equation, we can rearrange it to solve for y:
x - 5y = 7
-5y = -x + 7
y = (1/5)x - 7/5
The second equation is 3x-2y=-4. To graph this equation, we can rearrange it to solve for y:
3x - 2y = -4
-2y = -3x - 4
y = (3/2)x + 2
Now we can plot the graphs of both equations on the same coordinate system:
Equation 1: y = (1/5)x - 7/5
Equation 2: y = (3/2)x + 2
After plotting the equations, we find that the lines intersect at the point (3, -2). Therefore, the solution to both equations is x = 3 and y = -2.