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Solve Systems of
Linear Equations

Master three methods to find where two lines meet: graphing, substitution, and elimination. Every solution is an ordered pair \((x, y)\).

8th Grade Math

What Is a System of Equations?

A system of linear equations is a set of two equations with two unknowns, \(x\) and \(y\). We need to find the one pair of values that makes both equations true at the same time.

The Intersection Point

Each equation is a line on the coordinate plane. The solution is the point where the two lines cross.

x y 1 2 3 4 1 2 3 4 (1, 3) y = 2x + 1 y = -x + 4

Three Possible Outcomes

One Solution
Lines intersect at exactly one point. Different slopes. The system is consistent and independent.
No Solution
Lines are parallel: same slope, different intercepts. They never cross. The system is inconsistent.
Infinitely Many
Lines are identical: same slope and same intercept. Every point on the line is a solution. Dependent system.

Method 1: Graphing

Graph both equations on the same coordinate plane and read the intersection point. This gives an estimate of the solution.

1
Rewrite each equation in slope-intercept form: \(y = mx + b\).
2
Plot the y-intercept \(b\), then use the slope \(m\) to find more points.
3
Draw both lines and identify where they cross.
4
Read the coordinates of the intersection and verify in both equations.

Method 2: Substitution

Substitution works by solving one equation for a variable, then plugging that expression into the other equation.

1
Isolate one variable in either equation (e.g., solve for \(y\)).
2
Substitute that expression into the other equation.
3
Solve the resulting one-variable equation.
4
Back-substitute to find the other variable, then check.

Method 3: Elimination

Elimination (also called the addition method) adds or subtracts the equations to cancel one variable.

1
Line up both equations in standard form: \(Ax + By = C\).
2
Multiply one or both equations so that the coefficients of one variable are opposites.
3
Add the equations to eliminate that variable.
4
Solve for the remaining variable, then back-substitute.

Substitution vs. Elimination

Substitution Elimination
Best when A variable is already isolated or has coefficient 1 Both equations are in standard form
Key step Replace variable with expression Add/subtract equations
Watch for Distributing correctly after substituting Multiplying entire equation (both sides)
Example cue \(y = 3x - 5\) \(2x + 3y = 7\)

Choosing Your Strategy

1
Check by inspection first: Do the equations simplify to the same line, or are they parallel?
2
Is a variable already isolated? If so, use substitution.
3
Are coefficients easy to match? If so, use elimination.
4
Always verify your solution by plugging \((x, y)\) back into both original equations.

Example 1: Substitution

1
Substitute the first equation into the second: \(2x + (3x - 2) = 8\).
2
Combine like terms: \(5x - 2 = 8\), so \(5x = 10\), giving \(x = 2\).
3
Back-substitute: \(y = 3(2) - 2 = 4\).
4
Check: \(2(2) + 4 = 8\) ✓ — Solution: \((2, 4)\).

Example 2: Elimination

1
Notice: the \(y\)-coefficients are already opposites (\(+2y\) and \(-2y\)).
2
Add the equations: \(3x + 2y + x - 2y = 16 + (-4)\) gives \(4x = 12\).
3
Solve: \(x = 3\). Substitute back: \(3 - 2y = -4\), so \(-2y = -7\), \(y = 3.5\).
4
Check: \(3(3) + 2(3.5) = 9 + 7 = 16\) ✓ — Solution: \((3,\; 3.5)\).

Example 3: Graphing Method

1 2 3 4 5 -1 -2 1 2 3 4 5 x y (2, 3) y = x + 1 y = -2x + 7

Solving by Inspection

Sometimes you can determine the number of solutions without solving by comparing slopes and intercepts.

No Solution
\( y = 2x + 3 \)
\( y = 2x - 1 \)

Same slope (\(m = 2\)), different intercepts. Parallel lines never meet.
Infinite Solutions
\( y = 2x + 3 \)
\( 4x - 2y = -6 \)

The second equation simplifies to \(y = 2x + 3\). Same line: every point works.

Example 4: No Solution Algebraically

1
Multiply the first equation by \(-2\): \(-4x - 2y = -10\).
2
Add to the second: \(-4x - 2y + 4x + 2y = -10 + 3\).
3
This gives \(0 = -7\), which is a false statement.
4
A contradiction means the lines are parallel: no solution.

Common Mistakes to Avoid

✕ Forgetting to Distribute
When substituting \(y = 3x - 2\) into \(2(3x - 2)\), you must distribute to get \(6x - 4\), not \(6x - 2\).
✕ Multiplying Only One Side
In elimination, you must multiply every term in the equation, including the constant on the right side.
✕ Not Checking the Answer
Always plug \((x, y)\) back into both original equations. Checking only one equation can miss errors.

Method Summary

Graphing Substitution Elimination
Precision Estimated Exact Exact
Best for Visualizing Variable already isolated Standard form equations
Speed Slow Medium Fast (when coefficients match)
Detects special cases? Parallel / same line visible Contradiction / identity Contradiction / identity

Question 1 of 5

Solve the system by substitution:
\( y = x + 3 \)
\( 2x + y = 12 \)

Question 2 of 5

Solve the system by elimination:
\( 3x + y = 10 \)
\( 3x - y = 2 \)

Question 3 of 5

How many solutions does this system have?
\( y = -4x + 5 \)
\( y = -4x + 1 \)

Question 4 of 5

Solve the system:
\( 5x - 2y = 1 \)
\( x + 2y = 11 \)

Question 5 of 5

Which method would be most efficient for solving:
\( y = 7x - 10 \)
\( 3x + 2y = 4 \)