Lesson 4 • 45 min read
Systems of Equations & Inequalities
In This Lesson
1What Are Systems of Equations?
A system of equations is a set of two or more equations with the same variables. The solution is the point(s) where all equations are true simultaneously.
Example System:
x + y = 5
x - y = 1
Solution: x = 3, y = 2 (satisfies both equations)
One Solution
Lines intersect at one point
Most common case
No Solution
Lines are parallel
Same slope, different intercepts
Infinite Solutions
Lines are the same
Equations are equivalent
2Substitution Method
Steps
- Solve one equation for one variable
- Substitute that expression into the other equation
- Solve for the remaining variable
- Substitute back to find the other variable
- Check your answer in both original equations
Example: Solve the system
y = 2x - 1 ← Already solved for y
3x + y = 9
Step 1: Substitute y = 2x - 1 into second equation
3x + (2x - 1) = 9
Step 2: Solve for x
5x - 1 = 9
5x = 10
x = 2
Step 3: Substitute back to find y
y = 2(2) - 1 = 3
Solution: (2, 3)
Example 2: Solve the system
x + 2y = 7
3x - y = 7
Step 1: Solve first equation for x
x = 7 - 2y
Step 2: Substitute into second equation
3(7 - 2y) - y = 7
21 - 6y - y = 7
21 - 7y = 7
-7y = -14
y = 2
Step 3: Substitute back
x = 7 - 2(2) = 3
Solution: (3, 2)
3Elimination Method
Steps
- Arrange equations with variables aligned
- Multiply one or both equations to get opposite coefficients
- Add equations to eliminate one variable
- Solve for the remaining variable
- Substitute back to find the other variable
Example 1: Opposite coefficients already
x + y = 5
x - y = 1
Add equations: y terms cancel!
2x = 6
x = 3
Substitute: 3 + y = 5
y = 2
Solution: (3, 2)
Example 2: Need to multiply first
2x + 3y = 12
4x - y = 5
Multiply second by 3:
2x + 3y = 12
12x - 3y = 15
Add:
14x = 27
x = 27/14
Substitute to find y...
When to Use Each Method
Substitution: When one variable is already isolated (y = ... or x = ...)
Elimination: When coefficients are easy to match, or both equations in standard form
4Graphing Method
Steps
- Convert both equations to slope-intercept form (y = mx + b)
- Graph both lines on the same coordinate plane
- Find the point of intersection (the solution)
- Verify by substituting into both original equations
Example: Solve by graphing
y = x + 1
y = -x + 3
Line 1: slope = 1, y-intercept = 1
Line 2: slope = -1, y-intercept = 3
Intersection point: (1, 2)
One Solution
Lines cross at one point
Different slopes
No Solution
Lines never cross
Parallel (same slope)
Infinite Solutions
Lines overlap
Same line
5Linear Inequalities
Inequality Symbols
<
Less than
>
Greater than
≤
Less than or equal
≥
Greater than or equal
⚠️ Important Rule
When multiplying or dividing by a NEGATIVE number, you must FLIP the inequality sign!
-2x > 6 → x < -3 (flipped!)
Solving Inequalities
3x - 5 > 7
3x > 12 ← Add 5
x > 4 ← Divide by 3
Solution: all numbers greater than 4
Graphing on Number Line
Open circle (○): < or > (not included)
Closed circle (●): ≤ or ≥ (included)
x > 4 → ○────────→
x ≤ 4 → ←────────●
Graphing Linear Inequalities (Two Variables)
- Graph the boundary line (use = instead of inequality)
- Use dashed line for < or >, solid line for ≤ or ≥
- Pick a test point (like 0,0) not on the line
- Shade the side that makes the inequality true
Example: y > 2x - 1
Dashed line (not equal), shade above (y greater)