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Solve the system of equations using elimination and identify the solution. List your answers alphabetically in the ordered pairs.

[tex]\[
\begin{cases}
5e + 4f = 9 \\
4e + 5f = 9
\end{cases}
\][/tex]

A. [tex]\((-1, -1)\)[/tex]

B. [tex]\((1, 1)\)[/tex]

C. [tex]\((-2, -2)\)[/tex]

D. [tex]\((2, 2)\)[/tex]

E. No Solution

F. Infinitely Many Solutions


Sagot :

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].