make simultaneous equations easy whether using elimination or subsitution method show both make the concept clear and concise

Let's simplify the two main methods for solving simultaneous equations: **elimination** and **substitution**. We'll break each one down step-by-step.

### Example Problem
Suppose we have these two equations:
\[
\begin{cases} 
2x + 3y = 13 \\
x - y = 1
\end{cases}
\]

### Method 1: Elimination
The elimination method involves combining the equations to eliminate one of the variables.

1. **Align the Equations**: Write both equations with the variables aligned:
   \[
   2x + 3y = 13 \quad \text{(Equation 1)}
   \]
   \[
   x - y = 1 \quad \text{(Equation 2)}
   \]

2. **Make the Coefficients of \(x\) or \(y\) the Same**: 
   - Here, we can multiply **Equation 2** by 2 so that both equations have \(2x\):
     \[
     2(x - y) = 2 \times 1
     \]
     This gives us:
     \[
     2x - 2y = 2 \quad \text{(New Equation 2)}
     \]

3. **Subtract the Equations to Eliminate \(x\)**:
   \[
   (2x + 3y) - (2x - 2y) = 13 - 2
   \]
   Simplifying this gives:
   \[
   5y = 11
   \]

4. **Solve for \(y\)**:
   \[
   y = \frac{11}{5} = 2.2
   \]

5. **Substitute \(y\) Back to Find \(x\)**:
   Substitute \(y = 2.2\) into **Equation 2**:
   \[
   x - 2.2 = 1
   \]
   \[
   x = 1 + 2.2 = 3.2
   \]

So, the solution is:
\[
x = 3.2, \quad y = 2.2
\]

### Method 2: Substitution
In the substitution method, solve one equation for one variable and substitute it into the other equation.

1. **Solve for One Variable in Terms of the Other**:
   - From **Equation 2**, we can solve for \(x\) in terms of \(y\):
     \[
     x = y + 1
     \]

2. **Substitute \(x\) in the Other Equation**:
   Substitute \(x = y + 1\) into **Equation 1**:
   \[
   2(y + 1) + 3y = 13
   \]

3. **Expand and Simplify**:
   \[
   2y + 2 + 3y = 13
   \]
   Combine like terms:
   \[
   5y + 2 = 13
   \]
   \[
   5y = 11
   \]
   \[
   y = 2.2
   \]

4. **Substitute \(y\) Back to Find \(x\)**:
   Substitute \(y = 2.2\) back into \(x = y + 1\):
   \[
   x = 2.2 + 1 = 3.2
   \]

The solution is the same:
\[
x = 3.2, \quad y = 2.2
\]

### Summary of Both Methods
- **Elimination**: Align equations, manipulate to match coefficients, and add/subtract to eliminate a variable.
- **Substitution**: Solve one equation for one variable, substitute into the other equation, and solve.

Would you like more practice with simultaneous equations or help with specific examples?

Here are five questions for additional practice:
1. How would you solve \(3x + 4y = 10\) and \(2x - y = 3\) using elimination?
2. What are the steps for solving \(x + 2y = 7\) and \(2x - y = 4\) by substitution?
3. How can you check if your solution to simultaneous equations is correct?
4. What happens if both equations in elimination result in a false statement (like \(0 = 5\))?
5. How would you adjust the substitution method if the equations are non-linear?

**Tip:** Choose elimination if the variables have coefficients that are easy to match; use substitution if isolating one variable is straightforward.

make simultaneous equations easy whether using elimination or substitution method show both make the concept clear and concise

Understand how to solve simultaneous equations with a clear guide to the elimination and substitution methods. Step-by-step examples make these algebraic techniques easy, helping students build problem-solving skills in Grades 8-10.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation