Solve each system of eaquations by graphing or using a table.
5x-y=12
3x+2y=12
so, draw the graphs, or make a table. No problema, right?
To solve this system of equations by graphing, we can plot the lines for each equation on a coordinate plane and find the point of intersection, which represents the solution.
1. First, let's solve the first equation, 5x - y = 12, for y:
Start by isolating the variable y:
-y = -5x + 12
Multiply both sides by -1 to have y positive:
y = 5x - 12
2. Now, solve the second equation, 3x + 2y = 12, for y:
Start by isolating the variable y:
2y = -3x + 12
Divide both sides by 2 to solve for y:
y = -1.5x + 6
3. Graph the lines:
Plot the points for each equation to draw the lines on a coordinate plane.
For the equation y = 5x - 12:
- Choose two values for x and substitute it into the equation to find the corresponding y-coordinates.
Let's choose x = 0:
y = 5(0) - 12 = -12
So, one point is (0, -12).
Let's choose x = 2:
y = 5(2) - 12 = -2
So, another point is (2, -2).
- Plot these two points and draw the line passing through them.
For the equation y = -1.5x + 6:
- Choose two values for x and substitute it into the equation to find the corresponding y-coordinates.
Let's choose x = 0:
y = -1.5(0) + 6 = 6
So, one point is (0, 6).
Let's choose x = 2:
y = -1.5(2) + 6 = 3
So, another point is (2, 3).
- Plot these two points and draw the line passing through them.
4. Find the point of intersection:
Look at the graphed lines and find the point where they intersect. This point represents the solution to the system of equations.
Alternatively, you can also solve this system of equations using a table:
1. Solve the first equation, 5x - y = 12, for y:
Substitute different values of x into the equation and solve for the corresponding y-values.
Let's choose x = -2, 0, and 2, for example:
- When x = -2:
5(-2) - y = 12
-10 - y = 12
-y = 12 + 10
-y = 22
y = -22
- When x = 0:
5(0) - y = 12
-y = 12
y = -12
- When x = 2:
5(2) - y = 12
10 - y = 12
-y = 2
y = -2
2. Solve the second equation, 3x + 2y = 12, for y:
Substitute different values of x into the equation and solve for the corresponding y-values.
Using the same x-values as before:
- When x = -2:
3(-2) + 2y = 12
-6 + 2y = 12
2y = 18
y = 9
- When x = 0:
3(0) + 2y = 12
2y = 12
y = 6
- When x = 2:
3(2) + 2y = 12
6 + 2y = 12
2y = 6
y = 3
3. Create a table with the x-values and their corresponding y-values for each equation.
First equation:
| x | y |
| -2 | -22 |
| 0 | -12 |
| 2 | -2 |
Second equation:
| x | y |
| -2 | 9 |
| 0 | 6 |
| 2 | 3 |
4. Compare the y-values in the table:
Look for the row where the y-values are equal for both equations. That row represents the point of intersection, which is the solution to the system of equations.
In this case, the point (2, 3) is the solution.
So, the solution to the system of equations is x = 2 and y = 3.