Now give the solution in terms of a and b to  2 4 1 1   x y  =  a b  .

The given system of equations is represented in matrix form as:

\[
\begin{pmatrix}
2 & 4 \\
1 & 1
\end{pmatrix}
\begin{pmatrix}
x \\
y
\end{pmatrix}
=
\begin{pmatrix}
a \\
b
\end{pmatrix}
\]

To solve for \( x \) and \( y \), we can express the system of equations as:

1. \( 2x + 4y = a \)
2. \( x + y = b \)

### Step 1: Solve the second equation for \( x \):

\[
x + y = b \quad \Rightarrow \quad x = b - y
\]

### Step 2: Substitute \( x = b - y \) into the first equation:

\[
2(b - y) + 4y = a
\]
\[
2b - 2y + 4y = a
\]
\[
2b + 2y = a
\]
\[
2y = a - 2b
\]
\[
y = \frac{a - 2b}{2}
\]

### Step 3: Substitute \( y = \frac{a - 2b}{2} \) into \( x = b - y \):

\[
x = b - \frac{a - 2b}{2}
\]
\[
x = b - \frac{a}{2} + b
\]
\[
x = 2b - \frac{a}{2}
\]

### Final Solution:

\[
x = 2b - \frac{a}{2}, \quad y = \frac{a - 2b}{2}
\]

Would you like further details or have any questions?

---

Here are 5 related questions:

1. How do you invert a 2x2 matrix?
2. Can this method be extended to larger systems of equations?
3. What is the determinant of the matrix in this system?
4. How can you interpret the solution geometrically?
5. How does the system change if \( a = b \)?

**Tip:** Always verify the solution by substituting it back into the original equations!

Discover how to solve a system of linear equations represented by a matrix. This detailed step-by-step solution explains the use of substitution to find values for x and y in terms of a and b. Suitable for students in Grades 10-12 studying systems of equations and matrix algebra.

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