Systems of Linear Equations: Substitution and Elimination

A System of Linear Equations consists of two or more linear equations involving the same set of variables. The solution is the point \((x, y)\) where the lines intersect.

Solving Methods

1. Substitution Method:
Solve one equation for one variable (e.g., \(x = \dots\)) and substitute that expression into the other equation. Best when a variable is already isolated.

2. Elimination Method (Addition Method):
Add or subtract the equations to eliminate one variable. Multiply one or both equations by a constant first if necessary to get matching coefficients.

3. Graphing:
Graph both lines. The intersection point is the solution.

Step-by-Step Example (Elimination)

Solve the system:

$$ \begin{cases} 2x + 3y = 10 \\ 4x - y = 6 \end{cases} $$

Step 1: Align equations
The equations are already aligned.

Step 2: Modify to eliminate a variable
Let’s eliminate \(y\). Multiply the second equation by 3.

$$ \begin{cases} 2x + 3y = 10 \\ 3(4x - y) = 3(6) \rightarrow 12x - 3y = 18 \end{cases} $$

Step 3: Add the equations

$$ (2x + 3y) + (12x - 3y) = 10 + 18 $$

$$14x = 28$$

Step 4: Solve for the first variable

$$x = \frac{28}{14} = 2$$

Step 5: Solve for the second variable
Substitute \(x = 2\) into the original second equation:

$$4(2) - y = 6$$

$$8 - y = 6$$

$$-y = -2 \rightarrow y = 2$$

Solution: \((2, 2)\)

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Why reading isn’t enough

Systems of equations appear frequently in standardized tests and real-world applications. Choosing the right method (Substitution vs. Elimination) saves valuable time.

Weekzen helps you practice both methods efficiently. Our adaptive exercises ensure you are comfortable solving systems with integers, fractions, and decimals.