Review the graphs of a system of two linear equations in two
variables: x - 5y = 7 and
3х - 2y = -4. Find the solution to both equations.
(1 point)
The intersection point is
(3, -1).
To find the solution to a system of two linear equations in two variables, you need to find the point where the two lines intersect.
First, let's start by graphing the two equations:
Equation 1: x - 5y = 7
Equation 2: 3x - 2y = -4
To graph the first equation, we can rearrange it into slope-intercept form (y = mx + b), where m is the slope and b is the y-intercept:
x - 5y = 7
-5y = -x + 7
y = (1/5)x - 7/5
The slope of the first equation is 1/5, and the y-intercept is -7/5.
Now let's graph the second equation. We can also rearrange it into slope-intercept form:
3x - 2y = -4
-2y = -3x - 4
y = (3/2)x + 2
The slope of the second equation is 3/2, and the y-intercept is 2.
Now that we have the equations in slope-intercept form, let's plot the lines on a graph:
On the graph, the first equation (y = (1/5)x - 7/5) will have a slope of 1/5 and a y-intercept of -7/5. The second equation (y = (3/2)x + 2) will have a slope of 3/2 and a y-intercept of 2.
Plotting the lines, we find that they intersect at a single point. This point is the solution to the system of equations.
To find the exact coordinates of the intersection point, we can solve the system algebraically. One way to do this is by substitution or elimination.
Let's solve the system using the substitution method:
Substitute the expression for y from equation 1 into equation 2:
3x - 2((1/5)x - 7/5) = -4
3x - (2/5)x + 14/5 = -4
(15/5)x - (2/5)x + 14/5 = -4
(13/5)x + 14/5 = -4
(13/5)x = -4 - 14/5
(13/5)x = -20/5
(13/5)x = -4
x = -4 * (5/13)
x = -20/13
Now substitute the value of x back into equation 1 to find the value of y:
x - 5y = 7
(-20/13) - 5y = 7
-20 - 65y = 91
-65y = 91 + 20
-65y = 111
y = 111 / -65
y = -111 / 65
So, the solution to the system of equations is x = -20/13 and y = -111/65.
The intersection point is (-20/13, -111/65).
I hope this helps! Let me know if you have any further questions.
To find the solution to both equations, we can graph them and determine the point of intersection. Here are the steps to graph the system of equations and find the solution:
1. Start with the first equation, x - 5y = 7. To plot this equation on a graph, we rearrange it to the slope-intercept form: y = (1/5)x - 7/5. This equation represents a line with a slope of 1/5 and a y-intercept of -7/5.
2. Next, consider the second equation, 3x - 2y = -4. Rearrange it to slope-intercept form: y = (3/2)x + 2. This equation represents a line with a slope of 3/2 and a y-intercept of 2.
3. Plot the two lines on the same graph. Choose a suitable coordinate system and scale for convenience.
4. Locate the point where the two lines intersect. This point represents the solution to the system of equations.
5. Use the coordinates of the intersection point to express the solution. The x-coordinate of the intersection point is the solution to x, while the y-coordinate is the solution to y.
Reviewing the graphs and locating the intersection point, the solution to the system of equations x - 5y = 7 and 3x - 2y = -4 is the point of intersection.