https://r2.erweima.ai/mathbot/image/1725057446638dqffhqcu.jpg

To solve the system of linear equations using the method of reduction (also known as the elimination method), we proceed step by step:

### Given System:
\[
\begin{cases}
4m - 2n = 5 \quad \text{(Equation 1)} \\
-3m + 3n = -2 \quad \text{(Equation 2)}
\end{cases}
\]

### Step 1: Align the Equations
Align both equations to prepare for elimination:

\[
4m - 2n = 5 \quad \text{(Equation 1)}
\]
\[
-3m + 3n = -2 \quad \text{(Equation 2)}
\]

### Step 2: Eliminate One Variable
To eliminate one variable, we need to make the coefficients of either \(m\) or \(n\) the same (but with opposite signs) in both equations.

Let's eliminate \(n\). We can do this by multiplying the entire Equation 1 by 3, and the entire Equation 2 by 2, so that the coefficients of \(n\) in both equations are \(-6n\) and \(6n\), respectively.

\[
3 \times (4m - 2n) = 3 \times 5
\]
\[
12m - 6n = 15 \quad \text{(Equation 3)}
\]

\[
2 \times (-3m + 3n) = 2 \times -2
\]
\[
-6m + 6n = -4 \quad \text{(Equation 4)}
\]

### Step 3: Add or Subtract the Equations
Now add Equations 3 and 4 to eliminate \(n\):

\[
(12m - 6n) + (-6m + 6n) = 15 + (-4)
\]

This simplifies to:

\[
6m = 11
\]

### Step 4: Solve for \(m\)
Solve for \(m\) by dividing both sides by 6:

\[
m = \frac{11}{6}
\]

### Step 5: Substitute to Find \(n\)
Substitute \(m = \frac{11}{6}\) back into one of the original equations to solve for \(n\). Let's use Equation 1:

\[
4\left(\frac{11}{6}\right) - 2n = 5
\]

Simplify and solve for \(n\):

\[
\frac{44}{6} - 2n = 5
\]

\[
\frac{22}{3} - 2n = 5
\]

Convert 5 to a fraction:

\[
\frac{22}{3} - 2n = \frac{15}{3}
\]

Now, subtract \(\frac{22}{3}\) from both sides:

\[
-2n = \frac{15}{3} - \frac{22}{3} = \frac{-7}{3}
\]

Finally, solve for \(n\) by dividing both sides by -2:

\[
n = \frac{-7}{3} \times \frac{-1}{2} = \frac{7}{6}
\]

### Final Solution:
\[
m = \frac{11}{6}, \quad n = \frac{7}{6}
\]

Would you like further details on any of these steps?

---

Here are five related questions to further your understanding:

1. How would you solve the system using the substitution method?
2. What happens if the system has no solution? How can you recognize this?
3. Can you solve a similar system where both equations are already in the form \(Ax + By = C\)?
4. What does it mean for a system of linear equations to be dependent?
5. How would you check your solution by plugging it back into the original equations?

**Tip:** When using the elimination method, always double-check your multiplication and addition to avoid errors in your final solution.

Learn how to solve a system of linear equations using the method of elimination step by step. This detailed guide covers aligning equations, eliminating variables, and solving for unknowns with clear examples. Suitable for high school students and those preparing for algebra exams.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation