4.1Introduction to Systems of Equations
A system of equations is two or more equations with the same variables. The solution is the ordered pair that satisfies all equations simultaneously.
Why This Matters
Systems of equations model situations with multiple unknowns — like finding the price of two items given two total costs. This concept is fundamental in economics, engineering, and Physics, and it's a major topic in Algebra 2.
Workbook
Lesson, vocabulary, worked examples, and practice problems.
Essential Question
What does it mean for an ordered pair to be a solution to a system of equations, and how many solutions can a system have?
Lesson Overview
A system of equations is a set of two or more equations that share the same variables. Solving a system means finding the value(s) of the variables that make all equations true at the same time. The solution is written as an ordered pair (x, y). Depending on the lines, a system can have exactly one solution (lines intersect), no solution (parallel lines), or infinitely many solutions (same line).
Different slopes
1 solution
Consistent-Independent
Same slope, diff. intercept
No solution
Inconsistent
Same slope, same intercept
∞ solutions
Consistent-Dependent
One solution: lines intersect at (1,2)
No solution: parallel lines never meet
Infinite solutions: same line
Worked Examples
System: y=2x−1 and y=−x+2 → solution (1,1)
What is a System of Equations?
Two or more equations with the same variables.
A solution is an ordered pair (x, y) that satisfies ALL equations simultaneously.
On a graph, the solution is the intersection point.
Is (2, 5) a solution to the system: y = 2x + 1 and y = x + 3?
Substitute (2, 5) into Equation 1: 5 = 2(2) + 1 = 4 + 1 = 5 ✓
Substitute (2, 5) into Equation 2: 5 = 2 + 3 = 5 ✓
Both equations are satisfied → (2, 5) is a solution.
Is (1, 4) a solution to the system: y = 3x + 1 and y = x + 2?
Substitute (1, 4) into Equation 1: 4 = 3(1) + 1 = 4 ✓
Substitute (1, 4) into Equation 2: 4 = 1 + 2 = 3 ✗
Equation 2 is not satisfied → NOT a solution.
Classify the system: y = 2x + 3 and y = 2x − 1. How many solutions?
Slope of Eq 1: m = 2. Slope of Eq 2: m = 2. Same slope.
y-intercept of Eq 1: b = 3. y-intercept of Eq 2: b = −1. Different intercepts.
Same slope, different y-intercepts → parallel lines → no intersection.
Classify the system: y = −x + 4 and 2y = −2x + 8. How many solutions?
Simplify Equation 2: divide both sides by 2 → y = −x + 4.
Both equations are now identical: y = −x + 4.
Same line → every point on the line is a solution.
Classify the system: y = x + 1 and y = −x + 5. How many solutions?
Slope of Eq 1: m = 1. Slope of Eq 2: m = −1.
Different slopes → lines will intersect at exactly one point.
Guided Practice
Guided Practice Video: Introduction to Systems of Equations
Watch the guided practice walkthrough for introduction to systems of equations, then complete the problems below.
Video by Sang Real Math
Watch on YouTube ↗Answers are in the Answer Key section.
Is (3, 7) a solution to the system y = 2x + 1 and y = x + 4? Substitute and verify both equations.
Hint: Substitute both values into each equation and check if both sides are equal.
Is (0, 3) a solution to the system y = x + 3 and y = 2x + 3? Show your work.
Hint: Substitute (0, 3) into both equations. Does it satisfy both?
Classify: y = 4x − 2 and y = 4x + 5. How many solutions? What type of system?
Hint: Compare the slopes. If slopes are equal, compare y-intercepts.
Classify: y = x + 2 and 3y = 3x + 6. How many solutions? What type of system?
Hint: Divide the second equation by 3 first, then compare to the first equation.
Classify: y = 3x − 1 and y = −2x + 4. How many solutions? What type of system?
Hint: Different slopes mean the lines will cross at exactly one point.
Key Vocabulary
System of Equations
Two or more equations with the same variables considered together.
Example: y = 2x + 1 and y = x + 3 form a system.
Solution of a System
An ordered pair (x, y) that satisfies every equation in the system simultaneously.
Example: (2, 5) satisfies both y = 2x + 1 and y = x + 3.
Consistent System
A system that has at least one solution.
Example: Lines that intersect or are identical.
Inconsistent System
A system with no solution — the lines are parallel and never intersect.
Example: y = 2x + 3 and y = 2x − 1 (same slope, different intercepts).
Dependent System
A system with infinitely many solutions — both equations describe the same line.
Example: y = −x + 4 and 2y = −2x + 8 (same line).
Independent System
A consistent system with exactly one solution — the lines intersect at one point.
Example: y = x + 1 and y = −x + 5 (different slopes).
Interactive Practice — 5 Questions
Which ordered pair is a solution to the system y = x + 1 and y = 2x − 1?
A system of two equations has lines with the same slope and different y-intercepts. How many solutions does it have?
Which system is dependent (infinitely many solutions)?
Is (−2, 3) a solution to y = −x + 1 and y = 2x + 7?
What type of system has exactly one solution?
Independent Practice
Answers are in the Answer Key section.
Independent Practice
Is (1, 3) a solution to y = 2x + 1 and y = x + 2? Verify both equations.
Classify: y = 5x + 1 and y = 5x − 3. How many solutions? What type?
Classify: y = 2x + 4 and 2y = 4x + 8. How many solutions? What type?
Classify: y = x + 1 and y = −x + 3. How many solutions? What type?
Is (0, 0) a solution to y = 3x and y = −2x? Verify algebraically.
Common Mistakes
Checking the solution in only one equation instead of both.
A solution to a system must satisfy both equations simultaneously. Substitute into each one.
Writing the solution as two separate values — e.g., 'x = 2, y = 5' — instead of an ordered pair.
Always write the solution as an ordered pair: (2, 5).
Confusing a consistent system (one solution) with an inconsistent system (no solution) when lines look close together on a graph.
Two lines that are not parallel will always intersect at exactly one point. Parallel lines never intersect.
Forgetting to simplify equations before classifying — e.g., not dividing 2y = 4x + 8 by 2 before comparing.
Always rewrite equations in slope-intercept form (y = mx + b) before comparing slopes and intercepts.
Math Tips
To check a solution: substitute the ordered pair into both equations — it must satisfy both.
One solution: lines have different slopes — they cross at exactly one point.
No solution: lines have the same slope but different y-intercepts — they are parallel.
Infinitely many solutions: both equations simplify to the same line (same slope and y-intercept).
Always write the solution as an ordered pair: (x, y).