Sure, let's solve the system of linear equations step-by-step.
We are given the system of equations:
[tex]\[ \left\{\begin{array}{l}
4x - 5y = -2 \\
12x - 10y = 14 \\
\end{array}\right. \][/tex]
We will solve these equations to find the values of [tex]\( x \)[/tex] and [tex]\( y \)[/tex].
### Step 1: Observe the equations
First, let's take a closer look at the system of equations:
1. [tex]\( 4x - 5y = -2 \)[/tex]
2. [tex]\( 12x - 10y = 14 \)[/tex]
### Step 2: Simplify the second equation
Notice that the coefficients of [tex]\( x \)[/tex] and [tex]\( y \)[/tex] in the second equation are multiples of those in the first equation. So, let's simplify the second equation by dividing its terms by 2:
[tex]\[ \frac{12x - 10y}{2} = \frac{14}{2} \][/tex]
This simplifies to:
[tex]\[ 6x - 5y = 7 \][/tex]
### Step 3: Solve the system by elimination
Now we have the following system of equations:
1. [tex]\( 4x - 5y = -2 \)[/tex]
2. [tex]\( 6x - 5y = 7 \)[/tex]
We will eliminate [tex]\( y \)[/tex] by subtracting the first equation from the second equation. Let’s do that:
[tex]\[ (6x - 5y) - (4x - 5y) = 7 - (-2) \][/tex]
Simplifying the left-hand side and right-hand side separately:
[tex]\[ 6x - 5y - 4x + 5y = 7 + 2 \][/tex]
[tex]\[ 2x = 9 \][/tex]
### Step 4: Solve for [tex]\( x \)[/tex]
Now we solve for [tex]\( x \)[/tex]:
[tex]\[ x = \frac{9}{2} \][/tex]
### Step 5: Substitute [tex]\( x \)[/tex] back into one of the original equations
We will substitute [tex]\( x = \frac{9}{2} \)[/tex] back into the first equation to solve for [tex]\( y \)[/tex]:
[tex]\[ 4 \left(\frac{9}{2}\right) - 5y = -2 \][/tex]
Simplify:
[tex]\[ 4 \cdot \frac{9}{2} = 18 \][/tex]
So,
[tex]\[ 18 - 5y = -2 \][/tex]
### Step 6: Solve for [tex]\( y \)[/tex]
Isolate [tex]\( y \)[/tex] on one side:
[tex]\[ 18 - 5y = -2 \][/tex]
Subtract 18 from both sides:
[tex]\[ -5y = -2 - 18 \][/tex]
[tex]\[ -5y = -20 \][/tex]
Divide both sides by -5:
[tex]\[ y = \frac{20}{5} = 4 \][/tex]
### Step 7: Summary of the solution
We have found the solutions:
[tex]\[ x = \frac{9}{2} \][/tex]
[tex]\[ y = 4 \][/tex]
Therefore, the solution to the system of equations is:
[tex]\[ \left( \frac{9}{2}, 4 \right) \][/tex]
