To solve the given system of equations using the elimination method, we follow these steps:
Given equations:
[tex]\[ 5e + 4f = 9 \tag{1} \][/tex]
[tex]\[ 4e + 5f = 9 \tag{2} \][/tex]
### Step 1: Multiply the equations to align coefficients
To eliminate one of the variables, we will multiply each equation by coefficients so that the coefficients of either [tex]\(e\)[/tex] or [tex]\(f\)[/tex] are the same for both equations.
We can multiply equation (1) by 4 and equation (2) by 5:
[tex]\[
4(5e + 4f) = 4(9)
\][/tex]
[tex]\[
5(4e + 5f) = 5(9)
\][/tex]
This yields:
[tex]\[ 20e + 16f = 36 \tag{3} \][/tex]
[tex]\[ 20e + 25f = 45 \tag{4} \][/tex]
### Step 2: Subtract the equations to eliminate [tex]\(e\)[/tex]
Next, we subtract equation (3) from equation (4):
[tex]\[
(20e + 25f) - (20e + 16f) = 45 - 36
\][/tex]
This simplifies to:
[tex]\[
9f = 9
\][/tex]
### Step 3: Solve for [tex]\(f\)[/tex]
[tex]\[
f = \frac{9}{9}
\][/tex]
[tex]\[
f = 1
\][/tex]
### Step 4: Substitute [tex]\(f\)[/tex] back into one of the original equations to solve for [tex]\(e\)[/tex]
Substitute [tex]\(f = 1\)[/tex] into equation (1):
[tex]\[
5e + 4(1) = 9
\][/tex]
[tex]\[
5e + 4 = 9
\][/tex]
[tex]\[
5e = 9 - 4
\][/tex]
[tex]\[
5e = 5
\][/tex]
[tex]\[
e = \frac{5}{5}
\][/tex]
[tex]\[
e = 1
\][/tex]
### Step 5: Write the solution as an ordered pair
The solution to the system of equations is:
[tex]\[(e, f) = (1, 1)\][/tex]
### Step 6: Verify the solution
We can check our solution by substituting [tex]\(e = 1\)[/tex] and [tex]\(f = 1\)[/tex] into both original equations:
For equation (1):
[tex]\[
5(1) + 4(1) = 9
\][/tex]
[tex]\[
5 + 4 = 9
\][/tex]
[tex]\[
9 = 9 \quad \text{(True)}
\][/tex]
For equation (2):
[tex]\[
4(1) + 5(1) = 9
\][/tex]
[tex]\[
4 + 5 = 9
\][/tex]
[tex]\[
9 = 9 \quad \text{(True)}
\][/tex]
Since the solution [tex]\((1, 1)\)[/tex] satisfies both equations, it is the correct solution.
Thus, the solution to the system of equations is [tex]\((1, 1)\)[/tex].
