Question

9 which is the graph of the system below?
$4x + 5y = 15$
$2x - 3y = 6$

Explanation:

Step1: Rewrite first equation in slope - intercept form

The first equation is \(4x + 5y=15\). Solve for \(y\):
\(5y=-4x + 15\)
\(y =-\frac{4}{5}x+3\). The \(y\) - intercept is \(3\) and the slope is \(-\frac{4}{5}\).

Step2: Rewrite second equation in slope - intercept form

The second equation is \(2x-3y = 6\). Solve for \(y\):
\(-3y=-2x + 6\)
\(y=\frac{2}{3}x - 2\). The \(y\) - intercept is \(-2\) and the slope is \(\frac{2}{3}\).

Step3: Analyze the graphs based on intercepts and slopes

• For the line \(y =-\frac{4}{5}x + 3\), it should cross the \(y\) - axis at \((0,3)\) and have a negative slope.
• For the line \(y=\frac{2}{3}x-2\), it should cross the \(y\) - axis at \((0, - 2)\) and have a positive slope.

Looking at the graphs:

• Graph A: The first line (with negative slope) has a \(y\) - intercept of \(3\), and the second line (with positive slope) has a \(y\) - intercept of \(-2\), which matches our equations.
• Graph B: The positive - slope line has a \(y\) - intercept of \(3\) (incorrect, should be \(-2\)) and the negative - slope line has a \(y\) - intercept of \(-2\) (incorrect, should be \(3\)).
• Graph C: The positive - slope line has a \(y\) - intercept of \(-1\) (incorrect, should be \(-2\)) and the negative - slope line has a \(y\) - intercept of \(3\) (correct for the first line but the second line's intercept is wrong).
• Graph D: The two lines intersect at a point that does not match the solution of the system and the intercepts are incorrect.

Answer:

A