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.
