Q.If $x \begin{bmatrix} 2 \\ 3 \end{bmatrix} + y \begin{bmatrix} -1 \\ 1 \end{bmatrix} = \begin{bmatrix} 10 \\ 5 \end{bmatrix}$, find the values of $x$ and $y$.

Concept: Linear Combination — we solve a vector equation by equating components.

Write the given equation as two scalar equations:

Add the two equations to eliminate $y$:

Substitute $x = 3$ into the first equation: …

The problem gives you a linear combination of two vectors equaling a third vector. A linear combination just means you scale each vector by some number (here $x$ and $y$) and add them. When two vectors are not multiples of each other (they aren't, here), they form a basis for the plane — so any target vector can be uniquely expressed as a combination of them.

The vector equation

is really two scalar equations hiding inside one compact form. Each row gives you one equation.

1. Write the two equations. From the first row: $2x + (-1)y = 10$, i.e.

From the second row: $3x + 1 \cdot y = 5$, i.e.

2. Solve the system. The simplest way here is elimination: add the two equations.

The $y$ terms cancel: $5x = 15$, so $x = 3$.

3. Find $y$. Substitute $x = 3$ into either equation. Using $3x + y = 5$: …

Method: Turning a column-vector equation into scalar equations

A linear combination of column vectors set equal to another column vector is really several scalar equations stacked together — one per row. Split it, then solve the resulting system.

Steps

Step 1: Read off one equation per row

The $i$-th entry on the left must equal the $i$-th entry on the right. For $x\begin{bmatrix}2\\3\end{bmatrix}+y\begin{bmatrix}-1\\1\end{bmatrix}=\begin{bmatrix}10\\5\end{bmatrix}$ this gives $2x-y=10$ and $3x+y=5$. …

Common Mistakes

Mistake 1: Trying to "divide" the equation by a vector

Why it's wrong: vectors and matrices have no division, so you cannot isolate $x$ by dividing through by $\begin{bmatrix}2\\3\end{bmatrix}$. Correct approach: break the single vector equation into component (row-wise) scalar equations.

Mistake 2: Pairing a scalar with the wrong vector …