Answered

Westonci.ca is the Q&A platform that connects you with experts who provide accurate and detailed answers. Experience the ease of finding accurate answers to your questions from a knowledgeable community of professionals. Get quick and reliable solutions to your questions from a community of experienced experts on our platform.

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]
Thank you for your visit. We're committed to providing you with the best information available. Return anytime for more. Thank you for visiting. Our goal is to provide the most accurate answers for all your informational needs. Come back soon. Stay curious and keep coming back to Westonci.ca for answers to all your burning questions.