The goal here is to estimate the solution to the system of equations by graphing.
We are given two linear equations:
1. [tex]\(3x + 5y = 14\)[/tex]
2. [tex]\(6x - 4y = 9\)[/tex]
Let's follow the step-by-step process to find the estimated solution:
### Step 1: Convert each equation into slope-intercept form ([tex]\(y = mx + b\)[/tex]).
#### Equation 1: [tex]\(3x + 5y = 14\)[/tex]
Solve for [tex]\(y\)[/tex]:
[tex]\[5y = -3x + 14\][/tex]
[tex]\[y = -\frac{3}{5}x + \frac{14}{5}\][/tex]
The slope-intercept form of the first equation is [tex]\( y = -\frac{3}{5}x + \frac{14}{5} \)[/tex].
#### Equation 2: [tex]\(6x - 4y = 9\)[/tex]
Solve for [tex]\(y\)[/tex]:
[tex]\[ -4y = -6x + 9 \][/tex]
[tex]\[ y = \frac{6}{4}x - \frac{9}{4} \][/tex]
[tex]\[ y = \frac{3}{2}x - \frac{9}{4} \][/tex]
The slope-intercept form of the second equation is [tex]\( y = \frac{3}{2}x - \frac{9}{4} \)[/tex].
### Step 2: Graph the equations
Now we'll graph the two equations:
#### Graph 1:
Equation: [tex]\( y = -\frac{3}{5}x + \frac{14}{5} \)[/tex]
- The y-intercept ([tex]\(b\)[/tex]) is [tex]\(\frac{14}{5} = 2.8\)[/tex].
- The slope ([tex]\(m\)[/tex]) is [tex]\(-\frac{3}{5}\)[/tex].
Use the intercept to plot the first point [tex]\((0, 2.8)\)[/tex], and use the slope to find another point: if [tex]\(x\)[/tex] increases by 5 units, [tex]\(y\)[/tex] decreases by 3 units.
#### Graph 2:
Equation: [tex]\( y = \frac{3}{2}x - \frac{9}{4} \)[/tex]
- The y-intercept ([tex]\(b\)[/tex]) is [tex]\(-\frac{9}{4} = -2.25\)[/tex].
- The slope ([tex]\(m\)[/tex]) is [tex]\(\frac{3}{2}\)[/tex].
Use the intercept to plot the first point [tex]\((0, -2.25)\)[/tex], and use the slope to find another point: if [tex]\(x\)[/tex] increases by 2 units, [tex]\(y\)[/tex] increases by 3 units.
### Step 3: Estimate the Intersection Point from Graphs
When graphing these two lines, you seek the point where they intersect. Based on the given problem, let's compare the candidate answers to our intersection estimates:
A. [tex]\(\left( \frac{7}{3}, -\frac{7}{2} \right)\)[/tex]
B. [tex]\(\left( \frac{5}{2}, \frac{4}{3} \right)\)[/tex]
C. [tex]\(\left( \frac{4}{3}, \frac{5}{2} \right)\)[/tex]
D. [tex]\(\left( -\frac{5}{2}, -\frac{7}{2} \right)\)[/tex]
Comparing these points to the location of the intersection that we obtained through our calculations, the best fit would likely be:
[tex]\[ \left( \frac{5}{2}, \frac{4}{3} \right) \][/tex]
Thus, the correct answer is:
B. [tex]\(\left( \frac{5}{2}, \frac{4}{3} \right)\)[/tex]
