Grade 8 Mathematics
A SYSTEM OF EQUATIONS is two or more equations with the same variables. The SOLUTION is the ordered pair (x, y) that makes BOTH equations true at the same time!
On a graph, the solution is where the two lines INTERSECT (cross)!
GRAPHING METHOD:
1. Graph both equations
2. Find where they intersect
3. Write the intersection point
SUBSTITUTION METHOD:
1. Solve one equation for a variable
2. Substitute into the other equation
3. Solve, then find the other variable
Solve the system: y = x + 1 and y = -x + 5
Graph y = x + 1: Slope = 1, y-intercept = 1. Points: (0, 1), (1, 2), (2, 3)
Graph y = -x + 5: Slope = -1, y-intercept = 5. Points: (0, 5), (1, 4), (2, 3)
Find intersection: The lines cross at (2, 3)
Check: y = x + 1: 3 = 2 + 1 = 3 ✓ | y = -x + 5: 3 = -2 + 5 = 3 ✓ | Solution: (2, 3)
Solve: y = 2x - 1 and x + y = 5
Identify: The first equation already has y isolated: y = 2x - 1
Substitute: Replace y in the second equation with (2x - 1):
x + (2x - 1) = 5
Solve for x: x + 2x - 1 = 5 → 3x - 1 = 5 → 3x = 6 → x = 2
Find y: Substitute x = 2 into y = 2x - 1: y = 2(2) - 1 = 4 - 1 = 3
Solution: (2, 3). Check: y = 2(2) - 1 = 3 ✓ and 2 + 3 = 5 ✓
Is (4, 1) a solution to: 2x + y = 9 and x - y = 3?
Check equation 1: 2x + y = 9 → 2(4) + 1 = 8 + 1 = 9 ✓ TRUE!
Check equation 2: x - y = 3 → 4 - 1 = 3 ✓ TRUE!
Conclusion: Since (4, 1) makes BOTH equations true, YES, it IS a solution!
1. Is (2, 5) a solution to the system: y = 3x - 1 and y = x + 3?
Check equation 1: _______________ Check equation 2: _______________
Answer: YES / NO (circle one)
2. Solve by substitution: y = x + 4 and 2x + y = 10
Solution: ( , )
3. Look at the graph. What is the solution to the system?
Solution: ( , )
4. How many solutions does this system have? y = 2x + 3 and y = 2x - 1
(Hint: What do you notice about the slopes and y-intercepts?)
Answer: _____________ (One solution / No solution / Infinite solutions)