Solve this system of equations by graphing. First graph the equations, and then type the solution.

Question

y=
5
2
x+4
y=
1
2
x

Answer 1

To solve the system of equations by graphing, we first need to set up the equations properly.

1. First Equation:
   \[ y = \frac{5}{2}x + 4 \]

2. Second Equation:
   \[ y = \frac{1}{2}x \]

Step 1: Graph the Equations

For the first equation \(y = \frac{5}{2}x + 4\):

• The y-intercept (where \(x=0\)) is 4, so the first point is (0, 4).
• To find another point, you can set \(x = 2\): \[ y = \frac{5}{2}(2) + 4 = 5 + 4 = 9 \] Another point is (2, 9).

For the second equation \(y = \frac{1}{2}x\):

• The y-intercept (where \(x=0\)) is 0, so the first point is (0, 0).
• To find another point, you can set \(x = 2\): \[ y = \frac{1}{2}(2) = 1 \] Another point is (2, 1).

Step 2: Plot the Points and Draw the Lines

• For the first equation, plot the points (0, 4) and (2, 9) and draw a line through them.
• For the second equation, plot the points (0, 0) and (2, 1) and draw a line through them.

Step 3: Find the Point of Intersection

The solution to the system of equations is the point where the two lines intersect.

Step 4: Determine the Solution

From the graphs, you would visually check where the two graphs intersect. Let's calculate the intersection algebraically:

Set the equations equal to each other: \[ \frac{5}{2}x + 4 = \frac{1}{2}x \]

Now, solve for \(x\):

1. Subtract \(\frac{1}{2}x\) from both sides: \[ \frac{5}{2}x - \frac{1}{2}x + 4 = 0 \] \[ \frac{4}{2}x + 4 = 0 \quad \Rightarrow \quad 2x + 4 = 0 \]
2. Subtract 4: \[ 2x = -4 \]
3. Divide by 2: \[ x = -2 \]

Plug \(x = -2\) back into one of the original equations to find \(y\): \[ y = \frac{1}{2}(-2) = -1 \]

Final Solution

The solution to the system of equations is: \[ \boxed{(-2, -1)} \]

This point is where the two lines intersect on the graph.