To express vector $\mathbf{c} = (4, 7)$ as a linear combination of the vectors $\mathbf{a} = (2, 1)$ and $\mathbf{b} = (1, 3)$, we need to find the scalars $s$ and $t$ such that:

$\mathbf{c} = s \cdot \mathbf{a} + t \cdot \mathbf{b}$

This expands to:

$(4, 7) = s \cdot (2, 1) + t \cdot (1, 3)$

Which gives two simultaneous equations:

1. $2s + t = 4$
2. $s + 3t = 7$

Now, let's solve this system of equations.

Step 1: Solve for $t$ in terms of $s$

From the first equation:

$t = 4 - 2s$

Step 2: Substitute $t = 4 - 2s$ into the second equation

Substitute into the second equation $s + 3t = 7$:

$s + 3(4 - 2s) = 7$

Simplify the equation:

$s + 12 - 6s = 7$

$-5s + 12 = 7$

$-5s = 7 - 12$

$-5s = -5$

$s = 1$

Step 3: Find $t$

Substitute $s = 1$ into $t = 4 - 2s$:

$t = 4 - 2(1) = 4 - 2 = 2$

Final Answer:

The values of $s$ and $t$ are: $s = 1, \quad t = 2$

This means $\mathbf{c} = 1 \cdot \mathbf{a} + 2 \cdot \mathbf{b}$.

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Would you like more details or have any questions?

Here are 5 related questions:

1. How would the values of $s$ and $t$ change if the vector $\mathbf{c}$ were different?
2. Can every vector $\mathbf{c}$ be written as a linear combination of $\mathbf{a}$ and $\mathbf{b}$?
3. How can you verify the solution $s = 1$ and $t = 2$ geometrically?
4. What happens if vectors $\mathbf{a}$ and $\mathbf{b}$ are parallel?
5. How would this system change in three dimensions?

Tip: In solving systems of linear equations, substitution is helpful when one equation is easy to solve for one variable.