Question

solve the system of equations y = 3x - 5 and y = \frac{1}{3}x + 3 using a graphical method

Explanation:

Step1: Recall graphical - solution concept

The solution of the system of equations \(y = 3x - 5\) and \(y=\frac{1}{3}x + 3\) is the point of intersection of the two lines.

Step2: Set the two equations equal

Since \(y=y\), we set \(3x - 5=\frac{1}{3}x+3\).
First, multiply through by 3 to clear the fraction: \(9x-15 = x + 9\).

Step3: Solve for \(x\)

Subtract \(x\) from both sides: \(9x - x-15=x - x + 9\), which gives \(8x-15 = 9\).
Add 15 to both sides: \(8x-15 + 15=9 + 15\), so \(8x=24\).
Divide both sides by 8: \(x = 3\).

Step4: Solve for \(y\)

Substitute \(x = 3\) into \(y = 3x-5\), then \(y=3\times3 - 5=9 - 5=4\).
We can also substitute \(x = 3\) into \(y=\frac{1}{3}x + 3\), \(y=\frac{1}{3}\times3+3=1 + 3=4\).

Answer:

\((3,4)\)