Estimate the solution to the following system of equations by graphing.

[tex]
\begin{array}{l}
9x + 8y = 26 \\
3x + 2y = 9
\end{array}
[/tex]

A. [tex]\left(-\frac{10}{3}, -\frac{1}{2}\right)[/tex]
B. [tex]\left(\frac{8}{3}, -\frac{1}{2}\right)[/tex]
C. [tex]\left(\frac{8}{3}, \frac{1}{2}\right)[/tex]
D. [tex]\left(\frac{10}{3}, -\frac{1}{2}\right)[/tex]



Answer :

To solve the system of equations by graphing and estimate the solution, we can follow these steps:

1. Write each equation in slope-intercept form (y = mx + b) to easily plot the lines.

For the first equation: [tex]\(9x + 8y = 26\)[/tex]

- Solve for [tex]\(y\)[/tex]:
[tex]\[ 8y = -9x + 26 \][/tex]
[tex]\[ y = -\frac{9}{8}x + \frac{26}{8} \][/tex]
[tex]\[ y = -\frac{9}{8}x + \frac{13}{4} \][/tex]

For the second equation: [tex]\(3x + 2y = 9\)[/tex]

- Solve for [tex]\(y\)[/tex]:
[tex]\[ 2y = -3x + 9 \][/tex]
[tex]\[ y = -\frac{3}{2}x + \frac{9}{2} \][/tex]

2. Graph both equations on the same coordinate plane:

- For [tex]\(y = -\frac{9}{8}x + \frac{13}{4}\)[/tex]:
- The y-intercept is [tex]\( \frac{13}{4} \)[/tex] or 3.25.
- The slope is [tex]\(- \frac{9}{8}\)[/tex], meaning for every 8 units you move to the right on the x-axis, you move 9 units down on the y-axis.

- For [tex]\(y = -\frac{3}{2}x + \frac{9}{2}\)[/tex]:
- The y-intercept is [tex]\( \frac{9}{2} \)[/tex] or 4.5.
- The slope is [tex]\(- \frac{3}{2}\)[/tex], meaning for every 2 units you move to the right on the x-axis, you move 3 units down on the y-axis.

3. Estimate the point of intersection:
By plotting these lines accurately on graph paper or using a graphing tool, you would find where the two lines intersect. The correct solution will be the coordinates of this point of intersection.

4. Identify the correct solution among the choices:
According to the numerical result already derived, the exact solution to this system of equations is:
[tex]\[ \left( \frac{10}{3}, -\frac{1}{2} \right) \][/tex]
Hence, the correct answer is:
[tex]\[ \boxed{\left( \frac{10}{3}, -\frac{1}{2} \right)} \][/tex]

So, the intersection point and thus the solution to the system of equations [tex]\(9x + 8y = 26\)[/tex] and [tex]\(3x + 2y = 9\)[/tex] is indeed at the coordinates [tex]\(\left(\frac{10}{3},-\frac{1}{2}\right)\)[/tex], which corresponds to option D.