Question

solve the following system of inequalities graphically on the set of axes below. state the coordinates of a point in the solution set.
y ≤ -\frac{3}{5}x + 7
y < 2x - 6

Explanation:

Step1: Identify the slope - intercept form

The first inequality is $y\leq-\frac{3}{5}x + 7$, with slope $m_1=-\frac{3}{5}$ and y - intercept $b_1 = 7$. The second inequality is $y<2x - 6$, with slope $m_2 = 2$ and y - intercept $b_2=-6$.

Step2: Graph the first inequality

Graph the line $y =-\frac{3}{5}x + 7$. Since it is $y\leq-\frac{3}{5}x + 7$, the line is solid and we shade the region below the line.

Step3: Graph the second inequality

Graph the line $y = 2x-6$. Since it is $y<2x - 6$, the line is dashed and we shade the region below the line.

Step4: Find the intersection region

The solution set is the region where the shaded areas of the two inequalities overlap.

Step5: Find a point in the solution set

To find a point, we can solve the system of equations $y=-\frac{3}{5}x + 7$ and $y = 2x-6$ (ignoring the inequality signs for now). Set $-\frac{3}{5}x + 7=2x-6$.
First, multiply through by 5 to get $-3x + 35 = 10x-30$.
Add $3x$ to both sides: $35=10x + 3x-30$, so $35 = 13x-30$.
Add 30 to both sides: $65 = 13x$, then $x = 5$.
Substitute $x = 5$ into $y=2x-6$, we get $y=2\times5-6=4$.
We can also test a point in the overlapping region. Let's try $x = 4$.
For $y\leq-\frac{3}{5}x + 7$, when $x = 4$, $y\leq-\frac{3}{5}\times4 + 7=-\frac{12}{5}+7=\frac{-12 + 35}{5}=\frac{23}{5}=4.6$.
For $y<2x - 6$, when $x = 4$, $y<2\times4-6=2$. A point in the solution set is $(4,1)$ (since $1\leq4.6$ and $1<2$).

Answer:

A point in the solution set is $(4,1)$