Given the following System of Inequalities:
[tex]\begin{cases}7x+4y<16 \\ -5x+4y\ge12\end{cases}[/tex]You need to remember that the Slope-Intercept form of the equation of a line is:
[tex]y=mx+b[/tex]Where "m" is the slope and "b" is the y-intercept.
The steps to graph the System of Inequalities are:
1. In this case, you know that the first line is:
[tex]7x+4y=16[/tex]So you need to solve for "y" in order to write it in Slope-Intercept form:
[tex]\begin{gathered} 4y=-7x+16 \\ \\ y=\frac{-7}{4}x+\frac{16}{4} \\ \\ y=-\frac{7}{4}x+4 \end{gathered}[/tex]You can identify that the y-intercept is:
[tex]b_1=4[/tex]2. Since the value of "y" is zero when the line intersects the x-axis, you can substitute that value into the equation and solve for "x", in order to find the x-intercept:
[tex]\begin{gathered} 7x+4y=16 \\ 7x+4(0)=16 \\ 7x=16 \\ \\ x=\frac{16}{7}\approx2.286 \end{gathered}[/tex]3. Now you know that the first line passes through these points:
[tex](0,4);(2.286,0)[/tex]4. The second line is:
[tex]-5x+4y=12[/tex]So you can solve for "y" in order to write it in Slope-Intercept form:
[tex]\begin{gathered} 4y=5x+12 \\ \\ y=\frac{5}{4}x+\frac{12}{4} \\ \\ y=\frac{5}{4}x+3 \end{gathered}[/tex]Notice that the y-intercept is:
[tex]b_2=3[/tex]5. Knowing that "y" is zero when the line intersects the x-axis, you can substitute that value into the equation of the second line and solve for "x" to find the x-intercept:
[tex]\begin{gathered} -5x+4y=12 \\ -5x+4(0)=12 \\ -5x=12 \\ \\ x=\frac{12}{-5} \\ \\ x=-2.4 \end{gathered}[/tex]