Skill: systems-equations

Solving Linear Systems

Two lines, one meeting point. This skill is about strategy: solve by graphing, substitution, or elimination — but first, choose the most efficient method, and always know whether the system has one solution, none, or infinitely many.

1 Strategy first

A system is two equations sharing the same x and y. The solution is the ordered pair that satisfies both — the intersection point of the two lines. Before computing, choose your method.

Method picker

one variable isolated → substitution · coefficients aligned → elimination · both in y = mx + b → graphing

Substitution shines when an equation already says y = … or x = …. Elimination shines when adding or subtracting wipes out a variable. Graphing is for estimates and pictures.

One solution: lines cross. None: parallel lines never meet. Infinite: the same line twice.

Classification: compare slopes. Different slopes → exactly one solution. Same slope, different intercept → no solution. Same slope and intercept → infinitely many.

2 Worked examples

One per method, plus a classification.

Example 1 Substitution

Solve y = 2x − 5 and 3x + y = 10.

y is isolated — substitute: 3x + (2x − 5) = 10, so 5x = 15, x = 3. Then y = 2(3) − 5 = 1.

Solution: (3, 1). Check: 3(3) + 1 = 10.

Example 2 Elimination

Solve 2x + 3y = 13 and 2x − y = 1.

The x-coefficients match — subtract: 4y = 12, so y = 3. Then 2x + 9 = 13 gives x = 2.

Solution: (2, 3).

Example 3 Classification

Classify y = 2x + 1 and y = 2x − 4.

Same slope 2, different intercepts: the lines are parallel. No solution.

Example 4 Verify a candidate

Is (2, −1) the solution of x + y = 1 and 2x − y = 5?

2 + (−1) = 1 and 2(2) − (−1) = 5 — both hold. Yes.

3 Common mistakes

Three traps, each with the wrong version and the fix.

1. Solving for x but forgetting y

Wrong: “the solution is x = 3.” Right: a system solution is an ordered pair — finish the job: (3, 1).

2. Elimination arithmetic errors

Wrong: multiplying one term of the equation to align coefficients. Right: multiply the entire equation — every term, both sides.

3. Using a sledgehammer method

Wrong: graphing y = 2x − 5 and 3x + y = 10 to guess the point. Right: y is isolated → substitution is exact and fast.

4 Quick checks

Try these yourself, then reveal the answer.

Which method is most efficient for x = 3y − 1 and 2x + y = 12?

Substitution — x is already isolated.

Solve y = x + 4 and y = −2x + 1.

x + 4 = −2x + 1 gives x = −1, y = 3: (−1, 3).

Classify: 3x + 2y = 6 and 6x + 4y = 12.

The second is double the first — same line: infinitely many solutions.

5 Key points

Remember

Choose the method first: isolated variable → substitution; aligned coefficients → elimination; y = mx + b pair → graphing.
The solution is an ordered pair — find both coordinates.
Different slopes → one solution. Same slope, different intercept → none. Same line → infinite.
Elimination: multiply the whole equation.
Always verify by substituting into both original equations.

Key vocabulary

System of equations

Two or more equations sharing the same variables.

Substitution

Replacing a variable with its expression from another equation.

Elimination

Adding or subtracting equations to wipe out one variable.

Consistent / inconsistent

Consistent systems have at least one solution; inconsistent have none.

Dependent

Equations describing the same line — infinitely many solutions.

Practice systems Open the system solver Explore in the lab