The elimination method for solving systems of linear equations involves manipulating the equations to eliminate one of the variables, enabling you to solve for the other. Here’s a step-by-step guide:

Steps to Solve by Elimination

1. Align the Equations: Write the system of equations in standard form: $ax + by = c$ $dx + ey = f$

2. Make the Coefficients of One Variable Equal: Multiply one or both equations by appropriate constants so that the coefficients of either $x$ or $y$ are opposites (e.g., $+3x$ and $-3x$).

3. Add or Subtract the Equations: Add or subtract the equations to eliminate one variable. This step simplifies the system into a single equation with one variable.

4. Solve for the Remaining Variable: Solve the resulting single-variable equation.

5. Substitute Back to Find the Other Variable: Substitute the value found in step 4 into one of the original equations to solve for the second variable.

6. Verify the Solution: Plug the values of $x$ and $y$ into both original equations to ensure they satisfy the system.

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Example Problem

Solve the system: $2x + 3y = 12$ $4x - 3y = 6$

1. The coefficients of $y$ are already opposites ($+3y$ and $-3y$).

2. Add the equations: $(2x + 3y) + (4x - 3y) = 12 + 6$ $6x = 18$

3. Solve for $x$: $x = 3$

4. Substitute $x = 3$ into one of the original equations, say $2x + 3y = 12$: $2(3) + 3y = 12$ $6 + 3y = 12$ $3y = 6 \implies y = 2$

5. Solution: $x = 3$, $y = 2$.

6. Verify:

   • First equation: $2(3) + 3(2) = 6 + 6 = 12$ ✔
   • Second equation: $4(3) - 3(2) = 12 - 6 = 6$ ✔

The solution is $(x, y) = (3, 2)$.

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Do you want to try a similar problem or discuss any steps in more detail?

Here are 5 questions to expand on this topic:

1. How does elimination compare to substitution for solving systems of equations?
2. Can elimination work with systems of equations with more than two variables?
3. What happens if both variables are eliminated when using this method?
4. How do you decide whether to add or subtract equations during elimination?
5. Why might elimination be preferred over graphing in certain cases?

Tip: Always double-check your solution by substituting it back into the original equations.