Question

6 solve the system of equations below by graphing each line on the coordinate plane provided. then place a point where the solution of the system occurs. \\( x + 2y = 4 \\) \\( 3x + 2y = 8 \\)

Explanation:

Step1: Rewrite equations in slope - intercept form

For the equation \(x + 2y=4\), solve for \(y\):
\(2y=-x + 4\), so \(y=-\frac{1}{2}x + 2\).
For the equation \(3x + 2y = 8\), solve for \(y\):
\(2y=-3x + 8\), so \(y=-\frac{3}{2}x+4\).

Step2: Find intercepts for \(x + 2y = 4\)

• When \(x = 0\), \(0+2y = 4\), \(y = 2\). So the \(y\) - intercept is \((0,2)\).
• When \(y = 0\), \(x+2\times0 = 4\), \(x = 4\). So the \(x\) - intercept is \((4,0)\).

Step3: Find intercepts for \(3x + 2y = 8\)

• When \(x = 0\), \(3\times0+2y = 8\), \(y = 4\). So the \(y\) - intercept is \((0,4)\).
• When \(y = 0\), \(3x+2\times0 = 8\), \(x=\frac{8}{3}\approx2.67\). So the \(x\) - intercept is \((\frac{8}{3},0)\).

Step4: Graph the lines and find intersection

We can also solve the system by subtraction. Subtract the first equation \(x + 2y=4\) from the second equation \(3x + 2y = 8\):
\((3x + 2y)-(x + 2y)=8 - 4\)
\(3x+2y - x - 2y=4\)
\(2x=4\), so \(x = 2\).
Substitute \(x = 2\) into \(x + 2y=4\):
\(2+2y=4\)
\(2y=2\), so \(y = 1\).

Answer:

The solution of the system is the point \((2,1)\)