Sure, let's solve the system of linear equations step-by-step.
We have the system:
1. [tex]\( 6x - 5y = 8 \)[/tex]
2. [tex]\( 2x + 4y = 14 \)[/tex]
### Step 1: Write the equations
- First equation: [tex]\( 6x - 5y = 8 \)[/tex]
- Second equation: [tex]\( 2x + 4y = 14 \)[/tex]
### Step 2: Using the elimination method
In this method, we want to eliminate one of the variables by making the coefficients of one of the variables the same in both equations.
First, let's multiply the second equation by 3 so that the coefficients of [tex]\( x \)[/tex] in both equations will match:
[tex]\[
3(2x + 4y) = 3(14)
\][/tex]
This simplifies to:
[tex]\[
6x + 12y = 42
\][/tex]
### Step 3: Subtract the equations
Now, we have two equations:
1. [tex]\( 6x - 5y = 8 \)[/tex]
2. [tex]\( 6x + 12y = 42 \)[/tex]
Next, subtract the first equation from the second equation:
[tex]\[
(6x + 12y) - (6x - 5y) = 42 - 8
\][/tex]
This simplifies to:
[tex]\[
6x + 12y - 6x + 5y = 34
\][/tex]
[tex]\[
17y = 34
\][/tex]
Solving for [tex]\( y \)[/tex]:
[tex]\[
y = \frac{34}{17} = 2
\][/tex]
Now we know [tex]\( y = 2 \)[/tex].
### Step 4: Substitute [tex]\( y \)[/tex] back into one of the original equations
Let’s put [tex]\( y = 2 \)[/tex] into the first equation:
[tex]\[
6x - 5(2) = 8
\][/tex]
This simplifies to:
[tex]\[
6x - 10 = 8
\][/tex]
Solving for [tex]\( x \)[/tex]:
[tex]\[
6x = 18
\][/tex]
[tex]\[
x = \frac{18}{6} = 3
\][/tex]
So we have [tex]\( x = 3 \)[/tex].
### Solution
The solution to the system of equations is:
[tex]\[
x = 3
\][/tex]
[tex]\[
y = 2
\][/tex]
We can verify by substituting [tex]\( x = 3 \)[/tex] and [tex]\( y = 2 \)[/tex] back into both original equations to ensure both are satisfied.
[tex]\[
6(3) - 5(2) = 18 - 10 = 8 \quad \text{(True)}
\][/tex]
[tex]\[
2(3) + 4(2) = 6 + 8 = 14 \quad \text{(True)}
\][/tex]
Therefore, the solution is:
[tex]\[
(x, y) = (3, 2)
\][/tex]
