Question
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Tutor-Verified Answer
Answer
The solution to the system is:
- \( x = \frac{228}{43} \) (approximately 5.302)
- \( y = \frac{70}{43} \) (approximately 1.628)
Solution
It looks like you have the following system of linear equations:
**b)** \( 5x - 4y = 20 \)
**c)** \( 7x + 3y = 42 \)
Let's solve this system for the variables \( x \) and \( y \) using the **elimination method**.
### Step 1: Align the Equations
Write the equations one below the other for clarity:
\[
\begin{cases}
5x - 4y = 20 \quad \text{(Equation b)} \\
7x + 3y = 42 \quad \text{(Equation c)}
\end{cases}
\]
### Step 2: Make the Coefficients of \( y \) Opposites
To eliminate \( y \), we need the coefficients of \( y \) in both equations to be opposites.
- **Multiply Equation b by 3:**
\( 5x \times 3 - 4y \times 3 = 20 \times 3 \)
\( 15x - 12y = 60 \)
\(\quad\) **(Equation b1)**
- **Multiply Equation c by 4:**
\( 7x \times 4 + 3y \times 4 = 42 \times 4 \)
\( 28x + 12y = 168 \)
\(\quad\) **(Equation c1)**
### Step 3: Add the Modified Equations
Add **Equation b1** and **Equation c1** to eliminate \( y \):
\[
15x - 12y + 28x + 12y = 60 + 168
\]
\[
43x = 228
\]
\[
x = \frac{228}{43} = 5.\overline{3023} \quad \text{or} \quad 5 \frac{13}{43}
\]
### Step 4: Substitute \( x \) Back into One of the Original Equations
Let's use **Equation b** to find \( y \):
\[
5x - 4y = 20
\]
\[
5\left(\frac{228}{43}\right) - 4y = 20
\]
\[
\frac{1140}{43} - 4y = 20
\]
\[
-4y = 20 - \frac{1140}{43}
\]
\[
-4y = \frac{860}{43} - \frac{1140}{43}
\]
\[
-4y = \frac{-280}{43}
\]
\[
y = \frac{-280}{43} \div -4 = \frac{70}{43} = 1.\overline{6279} \quad \text{or} \quad 1 \frac{27}{43}
\]
### **Solution:**
\[
x = \frac{228}{43} = 5 \frac{13}{43} \quad \text{(approximately 5.302)}
\]
\[
y = \frac{70}{43} = 1 \frac{27}{43} \quad \text{(approximately 1.628)}
\]
---
**Final Answer:**
The solution to the system is
x = 228⁄43 (which is 5 13⁄43)
and
y = 70⁄43 (which is 1 27⁄43).
Reviewed and approved by the UpStudy tutoring team
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Simplify this solution Beyond the Answer
Did you know that both equations represent straight lines on a graph? The solution to this set of equations, also known as a system of equations, occurs where these two lines intersect. This intersection point represents the values of \(x\) and \(y\) that satisfy both equations simultaneously. When solving these equations, be mindful of common pitfalls! One mistake is forgetting to align terms correctly while combining or eliminating variables, especially in the substitution or elimination methods. A quick check of your work can prevent those pesky errors and lead you to the correct solution in no time!
