Let's solve the given system of linear equations step by step:
Given:
[tex]\[
\begin{cases}
4x + 7y = 34 \\
6x + 8y = 43
\end{cases}
\][/tex]
Step 1: Write the equations
1. [tex]\(4x + 7y = 34\)[/tex]
2. [tex]\(6x + 8y = 43\)[/tex]
Step 2: Solve one of the equations for one variable [tex]\(x\)[/tex] in terms of [tex]\(y\)[/tex] (or vice versa)
For convenience, let's solve the first equation for [tex]\(x\)[/tex]:
Rearranging equation 1:
[tex]\[ 4x = 34 - 7y \][/tex]
Divide both sides by 4:
[tex]\[ x = \frac{34 - 7y}{4} \][/tex]
Step 3: Substitute this expression for [tex]\(x\)[/tex] in the second equation
Replace [tex]\(x\)[/tex] in equation 2:
[tex]\[ 6\left( \frac{34 - 7y}{4} \right) + 8y = 43 \][/tex]
Step 4: Simplify and solve for [tex]\(y\)[/tex]
First, multiply everything inside the parenthesis by 6:
[tex]\[ \frac{6 \cdot (34 - 7y)}{4} + 8y = 43 \][/tex]
Which simplifies to:
[tex]\[ \frac{204 - 42y}{4} + 8y = 43 \][/tex]
Multiply through by 4 to clear the fraction:
[tex]\[ 204 - 42y + 32y = 172 \][/tex]
Combine like terms:
[tex]\[ 204 - 10y = 172 \][/tex]
Solve for [tex]\(y\)[/tex]:
[tex]\[ -10y = 172 - 204 \][/tex]
[tex]\[ -10y = -32 \][/tex]
[tex]\[ y = \frac{32}{10} \][/tex]
[tex]\[ y = 3.2 \][/tex]
Step 5: Substitute [tex]\(y\)[/tex] back into the expression for [tex]\(x\)[/tex]
Using the expression [tex]\(x = \frac{34 - 7y}{4}\)[/tex]:
[tex]\[ x = \frac{34 - 7(3.2)}{4} \][/tex]
Calculate [tex]\(34 - 7(3.2)\)[/tex]:
[tex]\[ 34 - 22.4 = 11.6 \][/tex]
Divide by 4:
[tex]\[ x = \frac{11.6}{4} \][/tex]
[tex]\[ x = 2.9 \][/tex]
Solution:
The values of [tex]\(x\)[/tex] and [tex]\(y\)[/tex] that satisfy the given system of equations are:
[tex]\[ x = 2.9 \][/tex]
[tex]\[ y = 3.2 \][/tex]
Hence, the solution to the given system of equations is:
[tex]\[ \left(x, y\right) = \left(\frac{29}{10}, \frac{16}{5}\right) \][/tex]
