It looks like you've provided a set of systems of linear equations and their corresponding answers. Let's go through each system step by step, solve them, and verify the given answers.
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Problem 1:
$$
\begin{cases}
2x - 9y = 1 \quad \text{(1)}\\
x - 4y = 1 \quad \text{(2)}
\end{cases}
$$
Step 1: Solve equation (2) for $ x $:
$$
x = 1 + 4y
$$
Step 2: Substitute into equation (1):
$$
2(1 + 4y) - 9y = 1 \\
2 + 8y - 9y = 1 \\
2 - y = 1 \\
y = 1
$$
Step 3: Plug $ y = 1 $ back into $ x = 1 + 4y $:
$$
x = 1 + 4(1) = 5
$$
✔ So, $ x = 5, y = 1 $
✔️ Matches the given answer.
---
Problem 2:
$$
\begin{cases}
2x + y = 0 \quad \text{(1)}\\
2x - y = 1 \quad \text{(2)}
\end{cases}
$$
Step 1: Add equations (1) and (2):
$$
(2x + y) + (2x - y) = 0 + 1 \\
4x = 1 \Rightarrow x = \frac{1}{4}
$$
Step 2: Plug $ x = \frac{1}{4} $ into equation (1):
$$
2\left(\frac{1}{4}\right) + y = 0 \Rightarrow \frac{1}{2} + y = 0 \Rightarrow y = -\frac{1}{2}
$$
✔ So, $ x = \frac{1}{4}, y = -\frac{1}{2} $
✔️ Matches the given answer.
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Problem 3:
$$
\begin{cases}
5r + 2s = 23 \quad \text{(1)}\\
4r + s = 19 \quad \text{(2)}
\end{cases}
$$
Step 1: Solve equation (2) for $ s $:
$$
s = 19 - 4r
$$
Step 2: Substitute into equation (1):
$$
5r + 2(19 - 4r) = 23 \\
5r + 38 - 8r = 23 \\
-3r = -15 \Rightarrow r = 5
$$
Step 3: Plug $ r = 5 $ into $ s = 19 - 4r $:
$$
s = 19 - 4(5) = 19 - 20 = -1
$$
✔ So, $ r = 5, s = -1 $
✔️ Matches the given answer.
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Problem 4:
$$
\begin{cases}
t - 4v = 1 \quad \text{(1)}\\
2t - 9v = 3 \quad \text{(2)}
\end{cases}
$$
Step 1: Solve equation (1) for $ t $:
$$
t = 1 + 4v
$$
Step 2: Substitute into equation (2):
$$
2(1 + 4v) - 9v = 3 \\
2 + 8v - 9v = 3 \\
2 - v = 3 \Rightarrow -v = 1 \Rightarrow v = -1
$$
Step 3: Plug $ v = -1 $ into $ t = 1 + 4v $:
$$
t = 1 + 4(-1) = 1 - 4 = -3
$$
✔ So, $ t = -3, v = -1 $
✔️ Matches the given answer.
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✔ Final Verification:
All four systems have been solved correctly, and the provided answers are accurate.
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✔ Summary of Solutions:
1. $ x = 5,\ y = 1 $
2. $ x = \frac{1}{4},\ y = -\frac{1}{2} $
3. $ r = 5,\ s = -1 $
4. $ t = -3,\ v = -1 $
All answers match the given solutions.
Let me know if you'd like these solved using matrices or elimination method instead!
Parent Tip: Review the logic above to help your child master the concept of substitution method worksheet.
