Graph the following system of equations.
y = 3x + 9
6x − 2y = 6
What is the solution to the system
To graph the system, we can first rearrange the equations into slope-intercept form, y = mx + b, where m is the slope and b is the y-intercept.
y = 3x + 9 can be written as y = 3x + 0 + 9, so the slope is 3 and the y-intercept is 9. We can plot this point on the y-axis.
6x - 2y = 6 can be simplified by dividing both sides by 2: 3x - y = 3. Solving for y, we get y = 3x - 3, which is already in slope-intercept form with slope 3 and y-intercept -3. We can plot this point on the y-axis.
Now we can use the slopes and intercepts to graph the lines and find the point where they intersect, which is the solution to the system.
Graph of the system:
To find the solution, we can see that the lines intersect at the point (2, 15), which is the solution to the system. Therefore, x = 2 and y = 15.
To graph the system of equations, we can use the slope-intercept form (y = mx + b) and the standard form (Ax + By = C).
First, let's rearrange the equations to match the form we need:
Equation 1: y = 3x + 9
Equation 2: 6x - 2y = 6
Now we can graph them step by step:
Step 1: Graph Equation 1
Since Equation 1 is already in slope-intercept form (y = mx + b), we can immediately identify the slope and the y-intercept.
Slope (m) = 3
y-intercept (b) = 9
To graph Equation 1:
1. Plot the y-intercept (point: (0, 9)).
2. Use the slope to find additional points. Since the slope is 3, you can count 3 units up and 1 unit to the right from the y-intercept. Plot that point (1, 12). Alternatively, you can go 3 units down and 1 unit to the left from the y-intercept. Plot that point (-1, 6).
3. Connect the points to draw a straight line.
Step 2: Graph Equation 2
Since Equation 2 is in standard form (Ax + By = C), let's rewrite it in slope-intercept form.
Start by isolating y in Equation 2:
6x - 2y = 6
-2y = -6x + 6
Divide both sides by -2:
y = 3x - 3
Now we can identify the slope and the y-intercept.
Slope (m) = 3 (same as Equation 1)
y-intercept (b) = -3
To graph Equation 2:
1. Plot the y-intercept (point: (0, -3)).
2. Use the slope to find additional points. Following the same process as for Equation 1, you can find the points (1, 0) and (-1, -6).
3. Connect the points to draw a straight line.
Step 3: Graphing the System of Equations
Now that we have graphed both equations, we can see where they intersect, which represents the solution to the system.
By inspecting the lines, we can see that they intersect at the point (0, 9).
Therefore, the solution to the system of equations is x = 0 and y = 9.