Question

Solving systems by elimination 7x-4y=27

Answer 1

To solve the equation \( 7x - 4y = 27 \) by elimination, we typically need a second equation. Since you provided only one equation, we can create a system by introducing another equation.

Let's use the equation \( 2x + 3y = 12 \) as our second equation. Now we have the system:

1. \( 7x - 4y = 27 \)
2. \( 2x + 3y = 12 \)

Now, we will solve this system using the elimination method.

### Step 1: Multiply the equations if necessary

We want to eliminate one variable by making their coefficients equal. Let's multiply the second equation by 4 so that the coefficients of \( y \) in both equations match.

\[
4(2x + 3y) = 4(12) \implies 8x + 12y = 48
\]

Now we have:
1. \( 7x - 4y = 27 \)
2. \( 8x + 12y = 48 \)

### Step 2: Eliminate a variable

We will try to eliminate \( y \) by multiplying the first equation by 3 and the second equation by 1 (no change is needed).

\[
3(7x - 4y) = 3(27) \implies 21x - 12y = 81
\]

Now our system looks like this:
1. \( 21x - 12y = 81 \)
2. \( 8x + 12y = 48 \)

### Step 3: Add the equations

Now let's add the two equations together to eliminate \( y \):

\[
(21x - 12y) + (8x + 12y) = 81 + 48
\]

This simplifies to:

\[
29x = 129
\]

### Step 4: Solve for \( x \)

Now, divide both sides by 29:

\[
x = \frac{129}{29} = 4.45
\]

### Step 5: Substitute \( x \) back to find \( y \)

Now, we will substitute \( x \) into one of the original equations, let's use the second equation:

\[
2(4.45) + 3y = 12
\]

This simplifies to:

\[
8.9 + 3y = 12
\]

Now, subtract 8.9 from both sides:

\[
3y = 12 - 8.9
\]

\[
3y = 3.1
\]

Now divide by 3:

\[
y = \frac{3.1}{3} \approx 1.03
\]

### Final answer

So the solution to the system of equations is:

\[
x \approx 4.45, \quad y \approx 1.03
\]

If you have a specific second equation to pair with \( 7x - 4y = 27 \), please provide it, and I can solve it directly!