To solve the system of linear equations
[tex]\[
\begin{array}{l}
3x + 2y = 10 \\
2x + 4y = 4
\end{array}
\][/tex]
we can use the method of solving systems of linear equations step-by-step, as follows:
1. Express the system of equations in matrix form:
We can represent the system as:
[tex]\[
A \vec{x} = \vec{b}
\][/tex]
where
[tex]\[
A = \begin{pmatrix}
3 & 2 \\
2 & 4
\end{pmatrix}
\][/tex]
and
[tex]\[
\vec{b} = \begin{pmatrix}
10 \\
4
\end{pmatrix}
\][/tex]
2. Find the inverse of the matrix \( A \) (if it exists):
However, here we approach solving using another reliable method, generally solving directly for x and y.
3. Find the solution using methods such as substitution or elimination:
Let's use elimination:
First, we can try to eliminate one of the variables by making the coefficients associated with one of the variables the same.
To eliminate \( y \), we can multiply the first equation by 2:
[tex]\[
6x + 4y = 20
\][/tex]
Now, subtract the second equation from this:
[tex]\[
(6x + 4y) - (2x + 4y) = 20 - 4
\][/tex]
Simplifying this:
[tex]\[
4x = 16
\][/tex]
Hence:
[tex]\[
x = 4
\][/tex]
4. Substitute the value of \( x \) back into one of the original equations to solve for \( y \):
Substitute \( x = 4 \) into the first equation:
[tex]\[
3(4) + 2y = 10
\][/tex]
Solving it:
[tex]\[
12 + 2y = 10
\][/tex]
[tex]\[
2y = 10 - 12
\][/tex]
[tex]\[
2y = -2
\][/tex]
[tex]\[
y = -1
\][/tex]
So, the solution to the system of equations is
[tex]\[
x = 4
\][/tex]
and
[tex]\[
y = -1.
\][/tex]
In conclusion, the solution to the system of equations
[tex]\[
\begin{array}{l}
3x + 2y = 10 \\
2x + 4y = 4
\end{array}
\][/tex]
is:
[tex]\[
x = 4
\][/tex]
[tex]\[
y = -1.
\][/tex]
