You can write the vector 𝑐⃗ as a linear combination of the vectors 𝑎⃗ and 𝑏$⃗ , like this

𝑐⃗ = 𝑠 ∗ 𝑎⃗ + 𝑡 ∗ 𝑏$⃗

Compute the values of s and t?

c = (4, 7)
a = (2, 1)
b = (1,3)

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} $.

---

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.

You can write the vector 𝑐⃗ as a linear combination of the vectors 𝑎⃗ and 𝑏$⃗ , like this 𝑐⃗ = 𝑠 ∗ 𝑎⃗ + 𝑡 ∗ 𝑏$⃗. Compute the values of s and t? c = (4, 7), a = (2, 1), b = (1,3)

Discover how to solve the system of equations to express the vector c = (4, 7) as a linear combination of vectors a = (2, 1) and b = (1, 3). This problem covers linear algebra concepts, including vector combinations and solving systems of linear equations. Suitable for students in Grades 9-12.

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