Question

llowing system of equations graphically on the set of
$y = \frac{1}{2}x + 7$
$5x + 2y = -10$

Explanation:

To solve the system of equations graphically, we first need to rewrite both equations in slope - intercept form (\(y = mx + b\), where \(m\) is the slope and \(b\) is the y - intercept) and then find two points for each line to plot them.

Step 1: Analyze the first equation

The first equation is already in slope - intercept form: \(y=\frac{1}{2}x + 7\)

• For the y - intercept, when \(x = 0\), \(y=\frac{1}{2}(0)+7=7\). So one point on this line is \((0,7)\).
• To find another point, we can let \(x = 2\) (we choose \(x = 2\) to make the calculation of \(y\) easy since the slope is \(\frac{1}{2}\)). Then \(y=\frac{1}{2}(2)+7=1 + 7=8\). So another point is \((2,8)\).

Step 2: Rewrite the second equation in slope - intercept form

We have the equation \(5x+2y=- 10\).

• Subtract \(5x\) from both sides: \(2y=-5x - 10\)
• Divide each term by 2: \(y=-\frac{5}{2}x-5\)
• For the y - intercept, when \(x = 0\), \(y=-\frac{5}{2}(0)-5=-5\). So one point on this line is \((0, - 5)\).
• To find another point, let \(x=-2\). Then \(y =-\frac{5}{2}(-2)-5=5 - 5 = 0\). So another point is \((-2,0)\).

Step 3: Graph the lines

• Plot the points \((0,7)\) and \((2,8)\) for the line \(y=\frac{1}{2}x + 7\) and draw a straight line through them.
• Plot the points \((0,-5)\) and \((-2,0)\) for the line \(y =-\frac{5}{2}x-5\) and draw a straight line through them.
• The point of intersection of the two lines is the solution of the system of equations.

To find the intersection algebraically (to verify the graphical solution):
Set \(\frac{1}{2}x + 7=-\frac{5}{2}x-5\)

• Add \(\frac{5}{2}x\) to both sides: \(\frac{1}{2}x+\frac{5}{2}x + 7=-\frac{5}{2}x+\frac{5}{2}x-5\)
• Combine like terms: \(\frac{1 + 5}{2}x+7=-5\), \(\frac{6}{2}x+7=-5\), \(3x+7=-5\)
• Subtract 7 from both sides: \(3x+7 - 7=-5 - 7\), \(3x=-12\)
• Divide both sides by 3: \(x=- 4\)

Substitute \(x = - 4\) into \(y=\frac{1}{2}x + 7\):
\(y=\frac{1}{2}(-4)+7=-2 + 7 = 5\)

The solution of the system of equations is \(x=-4\) and \(y = 5\), or the ordered pair \((-4,5)\).

Answer:

The solution of the system of equations is \(\boldsymbol{(-4,5)}\)