Solving systems of equations by elimination is a method used to find the values of variables that satisfy both equations simultaneously. Here are the steps to use the elimination method effectively:
Steps to Solve by Elimination
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Write the System of Equations: Start with the system of equations you want to solve. For example: \[ \begin{align*} 2x + 3y &= 16 \quad \text{(1)} \ 4x - 2y &= 8 \quad \text{(2)} \end{align*} \]
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Align the Equations: Make sure the equations are aligned with similar variables and constants in a straight line.
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Make the Coefficients of One Variable the Same: Choose one variable to eliminate. You may need to multiply one or both equations by a number to get the coefficients of that variable to match or to be opposites.
For these equations, let’s eliminate \(x\):
- Multiply equation (1) by 2: \[ 4x + 6y = 32 \quad \text{(3)} \]
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Subtract or Add the Equations: Now, subtract equation (2) from equation (3) to eliminate \(x\): \[ (4x + 6y) - (4x - 2y) = 32 - 8 \] Simplifying gives: \[ 8y = 24 \]
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Solve for the Remaining Variable: Divide both sides of the equation by 8: \[ y = 3 \]
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Substitute Back to Find the Other Variable: Now that you have the value of \(y\), substitute it back into one of the original equations to solve for \(x\). Using equation (1): \[ 2x + 3(3) = 16 \] \[ 2x + 9 = 16 \] \[ 2x = 7 \] \[ x = \frac{7}{2} \quad \text{or} \quad 3.5 \]
Summary of Results
The solution to the system of equations is: \[ x = \frac{7}{2}, \quad y = 3 \]
Additional Tips
- If both equations can be represented in a simpler form, do it before applying elimination.
- Sometimes, it might be easier to manipulate the equations to get negative coefficients for one variable, allowing that variable to be eliminated directly.
- Always check your solution by substituting both values back into the original equations.