Solve each system graphically.
4y = x + 8
3x - 2y = 6
Basically, what I have to do is find at least one point that works. I tried graphing it on graph paper, and it looked like about (1, 3.5). But when I tried placing in the equations, it did even work in the first one.
I do not have a graphing calculator, so don't suggest that. Do you have any methods to finding it, or can you help me find the answer?
Thanks so much! :-)
try around x = 4 and y = 4.5
4 y = x + 8
2 y = 3 x - 6
so
4 y = x + 8
4 y = 6 x - 12
so
x+8 = 6 x - 12
5 x = 20
x = 4
take it from there.
Silly me... we had just learned the elimination method and I had forgotten about it.
Thank you! :-)
To solve the system of equations graphically, you can follow these steps:
Step 1: Rearrange each equation to solve for y in terms of x.
Equation 1: 4y = x + 8
Divide both sides by 4: y = (1/4)x + 2
Equation 2: 3x - 2y = 6
Subtract 3x from both sides: -2y = -3x + 6
Divide both sides by -2: y = (3/2)x - 3
Now we have both equations in slope-intercept form (y = mx + b), where m is the slope and b is the y-intercept.
Step 2: Plot the lines on a graph. Since you don't have a graphing calculator, you can use graph paper or simply hand-draw the lines.
For Equation 1 (y = (1/4)x + 2), plot the y-intercept at (0, 2) and use the slope (1/4) to find a second point. You can choose any value for x and substitute it into the equation to find the corresponding y-value. For example, if we let x = 4, y = (1/4)(4) + 2 = 1 + 2 = 3. Thus, you can plot (4, 3).
For Equation 2 (y = (3/2)x - 3), plot the y-intercept at (0, -3) and find a second point similarly. Letting x = 2, y = (3/2)(2) - 3 = 3 - 3 = 0. Thus, you can plot (2, 0).
Step 3: Check the solution. Since you found the point (1, 3.5) by visually analyzing the graph, you need to verify if it satisfies both equations.
Substitute x = 1 and y = 3.5 into Equation 1:
4(3.5) = 1 + 8
14 = 9
The left-hand side does not equal the right-hand side, so (1, 3.5) is not a valid solution. You may have made an error when graphing or calculating.
Step 4: Find the intersection point. To determine the precise solution, you need to find the point where the two lines intersect. This is the solution to the system of equations.
Unfortunately, without a graphing calculator, finding the exact point of intersection may be challenging. You can use estimation techniques using the graph you drew.
Based on your estimation, you can try to narrow down the intersection point by looking at neighboring grid lines and estimating coordinates. For example, if you guessed the intersection is around (2, 1), try plugging these values into the original equations to see if they satisfy both equations.
Plug x = 2 and y = 1 into both equations:
For Equation 1: 4(1) = 2 + 8
4 = 10
Not a valid solution.
For Equation 2: 3(2) - 2(1) = 6
6 - 2 = 6
Valid solution!
From this estimation, it seems that the actual intersection point differs slightly from (2, 1), but you can refine your estimation by continuing this process or trying other nearby points.
This manual method might be a bit time-consuming, but without a graphing calculator, it is still a viable option to solve the system of equations.