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Solve the system of equations:

[tex]\[
\left\{
\begin{array}{l}
4x - 5y = -2 \\
12x - 10y = 14
\end{array}
\right.
\][/tex]

Sagot :

GinnyAnswer
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]
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