Answered

Westonci.ca is your go-to source for answers, with a community ready to provide accurate and timely information. Connect with a community of experts ready to provide precise solutions to your questions quickly and accurately. Explore comprehensive solutions to your questions from a wide range of professionals on our user-friendly 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]
Thanks for using our service. We're always here to provide accurate and up-to-date answers to all your queries. We appreciate your time. Please revisit us for more reliable answers to any questions you may have. Get the answers you need at Westonci.ca. Stay informed with our latest expert advice.