solve system of equations graphically and check
2x - y = 5
3x + 2y = 4
Do you know how to plot each equation onto the same graph?
After that, find where the two lines intersect and evaluate the values of x and y at the intersection point.
To solve the system of equations graphically, you need to find the points at which the two lines intersect on a graph.
Step 1: Convert the equations to slope-intercept form (y = mx + b).
Equation 1: 2x - y = 5
Rearrange the equation: -y = -2x + 5
Divide every term by -1: y = 2x - 5
Equation 2: 3x + 2y = 4
Rearrange the equation: 2y = -3x + 4
Divide every term by 2: y = (-3/2)x + 2
Step 2: Graph the two linear equations on the same coordinate system.
Plot the first equation, y = 2x - 5:
- Start by plotting the y-intercept, which is -5. This point is (0, -5).
- Then use the slope of 2 to find other points. For example, if you go one unit to the right (x + 1), you go two units up (y + 2). This gives another point as (1, -3).
- Connect the points with a straight line.
Plot the second equation, y = (-3/2)x + 2:
- Start by plotting the y-intercept, which is 2. This point is (0, 2).
- Then use the slope of -3/2 to find other points. For example, if you go two units to the right (x + 2), you go three units down (y - 3). This gives another point as (-2, 3).
- Connect the points with a straight line.
Step 3: Find the point of intersection.
The solution to the system of equations is the point where the two lines intersect on the graph. Find the coordinates of this point, which represents the values of x and y that satisfy both equations.
In this case, it seems difficult to determine the exact point of intersection by just looking at the graph. To ensure accuracy, we can use a graphing calculator or some software to find the coordinates of the intersection point.
Step 4: Check the solution.
To check if the solution is correct, substitute the values of x and y from the intersection point into both original equations and see if they hold true.
For our example, let's assume the intersection point is (2, -1). Plug these values into both equations:
Equation 1: 2x - y = 5
2(2) - (-1) = 5
4 + 1 = 5
5 = 5
Equation 2: 3x + 2y = 4
3(2) + 2(-1) = 4
6 - 2 = 4
4 = 4
Since both equations hold true with the values of x = 2 and y = -1, we can conclude that (2, -1) is the correct solution to the system of equations.