Solve each pair of simultaneous equations. Use your favourite algebraic method. $$ \begin{array}{lll}x+y=7 & ext { b } & y+x=19 \ ext { a } \ y=2 x-8 & & ext { c } \ & & 2 x+1\end{array} $$

Question

Solve each pair of simultaneous equations. Use your favourite algebraic method. $$ \begin{array}{lll}x+y=7 & 	ext { b } & y+x=19 \ 	ext { a } \ y=2 x-8 & & 	ext { c } \ & & 2 x+1\end{array} $$

UpStudy AI · Accepted Answer

Certainly! Let's solve each pair of simultaneous equations labeled **a** and **c** using substitution, which is an effective algebraic method.

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### **System a**

$$
\begin{cases}
x + y = 7 \quad \text{(1)} \
y = 2x - 8 \quad \text{(2)}
\end{cases}
$$

**Step 1:** Substitute equation (2) into equation (1):

$$
x + (2x - 8) = 7
$$

**Step 2:** Simplify and solve for $$ x $$:

$$
3x - 8 = 7 \
3x = 15 \
x = 5
$$

**Step 3:** Substitute $$ x = 5 $$ back into equation (2) to find $$ y $$:

$$
y = 2(5) - 8 \
y = 10 - 8 \
y = 2
$$

**Solution for System a:**
$$
x = 5, \quad y = 2
$$

---

### **System c**

$$
\begin{cases}
x + y = 19 \quad \text{(1)} \
y = 2x + 1 \quad \text{(2)}
\end{cases}
$$

**Step 1:** Substitute equation (2) into equation (1):

$$
x + (2x + 1) = 19
$$

**Step 2:** Simplify and solve for $$ x $$:

$$
3x + 1 = 19 \
3x = 18 \
x = 6
$$

**Step 3:** Substitute $$ x = 6 $$ back into equation (2) to find $$ y $$:

$$
y = 2(6) + 1 \
y = 12 + 1 \
y = 13
$$

**Solution for System c:**
$$
x = 6, \quad y = 13
$$

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**Summary of Solutions:**

- **System a:** $$ x = 5 $$, $$ y = 2 $$
- **System c:** $$ x = 6 $$, $$ y = 13 $$

If there was a **System b** intended, please provide the equations, and I'd be happy to help solve it as well!
Solutions for the systems are: - **System a:** $$ x = 5 $$, $$ y = 2 $$ - **System c:** $$ x = 6 $$, $$ y = 13 $$

Solve each pair of simultaneous equations.
Use your favourite algebraic method.

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