https://file.aiquickdraw.com/mathbot/file/172578512291373u5916g.jpg

this but with the coefficient:
if t = 0 then h(0) = 100

if t = 4 then h(4) = 338.4
16a + 4b + 100 = 338.4

if t = 10 then h(10) = 2
100a + 10b + 100 = 2

solve for the system (full solution)

Let's solve the system using the given data points and equations:

Given:

1. \( h(0) = 100 \)
2. \( h(4) = 338.4 \) → \( 16a + 4b + 100 = 338.4 \)
3. \( h(10) = 2 \) → \( 100a + 10b + 100 = 2 \)

These equations form a system of linear equations:

\[
16a + 4b + 100 = 338.4 \quad \text{(Equation 1)}
\]

\[
100a + 10b + 100 = 2 \quad \text{(Equation 2)}
\]

**Step 1: Simplify Equations**

Let's simplify the equations by isolating constants on one side:

**Equation 1:**
\[
16a + 4b = 338.4 - 100
\]

\[
16a + 4b = 238.4
\]

\[
4a + b = 59.6 \quad \text{(Divide by 4)}
\]

**Equation 2:**
\[
100a + 10b = 2 - 100
\]

\[
100a + 10b = -98
\]

\[
10a + b = -9.8 \quad \text{(Divide by 10)}
\]

Now, we have a simplified system:

\[
4a + b = 59.6 \quad \text{(Equation 3)}
\]

\[
10a + b = -9.8 \quad \text{(Equation 4)}
\]

**Step 2: Solve the System**

Subtract Equation 3 from Equation 4 to eliminate \( b \):

\[
(10a + b) - (4a + b) = -9.8 - 59.6
\]

\[
6a = -69.4
\]

\[
a = -\frac{69.4}{6}
\]

\[
a = -11.5667
\]

Now, substitute \( a = -11.5667 \) into Equation 3:

\[
4(-11.5667) + b = 59.6
\]

\[
-46.2668 + b = 59.6
\]

\[
b = 59.6 + 46.2668
\]

\[
b = 105.8668
\]

**Step 3: Verify the Solution**

To verify, plug \( a = -11.5667 \) and \( b = 105.8668 \) back into the original equations:

For Equation 1:
\[
16(-11.5667) + 4(105.8668) + 100 \approx 338.4
\]

For Equation 2:
\[
100(-11.5667) + 10(105.8668) + 100 \approx 2
\]

Both equations are satisfied.

**Solution:**
\[
a = -11.5667, \quad b = 105.8668, \quad c = 100
\]

Would you like a more detailed explanation or have any questions?

Here are some related questions you might be interested in:

1. How do you solve a system of equations with more than two variables?
2. What are some methods to solve systems of linear equations?
3. How can you verify the solution to a system of equations?
4. What if the system of equations has no solution?
5. How can graphing help in solving a system of equations?

**Tip:** Always double-check your simplified equations to ensure no errors were made during the simplification process.

Solve the system of equations given that:
1. h(0) = 100
2. h(4) = 338.4 → 16a + 4b + 100 = 338.4
3. h(10) = 2 → 100a + 10b + 100 = 2

Learn how to solve a system of linear equations involving height and time data points. Using substitution and elimination methods, this problem demonstrates the step-by-step process for solving for unknown coefficients in the equations. Suitable for students in Grades 9-11.

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