Explain, in complete sentences, how you would use the graphing method to solve the following system of equations. Provide the solution to the system and explain what the solution represents on the graph.
x + 4y = –16
3x + 2y = 12
To use the graphing method to solve the given system of equations, start by rewriting each equation in slope-intercept form, which is y = mx + b, where m represents the slope and b represents the y-intercept.
1) Begin with the equation x + 4y = -16:
- Subtract x from both sides to isolate the y term:
4y = -x - 16
- Divide both sides by 4 to solve for y:
y = (-1/4)x - 4
2) Move on to the equation 3x + 2y = 12:
- Subtract 3x from both sides to isolate the y term:
2y = -3x + 12
- Divide both sides by 2 to solve for y:
y = (-3/2)x + 6
Now that we have both equations in slope-intercept form, we can plot them on a graph.
- For the equation y = (-1/4)x - 4, the y-intercept is -4 (the point (0,-4)) and the slope is -1/4.
To visualize this on the graph, move one unit to the right and four units down to get the next point.
Plot these two points and draw a line passing through them.
- For the equation y = (-3/2)x + 6, the y-intercept is 6 (the point (0,6)) and the slope is -3/2.
To visualize this on the graph, move two units to the right and three units down to get the next point.
Plot these two points and draw a line passing through them.
Once both lines are plotted, examine the point(s) where they intersect. This point represents the solution to the system of equations. In this case, the intersection point is (4,-5).
On the graph, this solution represents the coordinates (x,y) where both equations are true simultaneously. So, the point (4,-5) satisfies both equations x + 4y = -16 and 3x + 2y = 12. This is the solution to the system of equations.
x+4y=-16 (1)
3x+2y=12 (2)
Multiply (1) by -3 to get the x's to cancel
-3x-12y=48 (3)
3x+2y=12 (2)
Now add (3) and (2) together.
-10y=60
Divide by -10.
y=-6
Substitude y=-6 into equation (1) or (2) to solve for x.