System of Linear Equations

Intuition — Both Conditions At Once

Line 1 Slope1

Line 1 y-intercept1

Line 2 Slope-1

Line 2 y-intercept3

👀 The Point Where Two Lines Meet

①A point on the first line satisfies y = ax + b

②A point on the second line satisfies y = cx + d

③Intersection: the unique (x, y) satisfying both

Elimination — Cancel One Unknown

📝 Elimination Example

①x + y = 5 …

①

②x - y = 1 …

②

③

① +

②: 2x = 6 → x = 3

④Substitute into

①: 3 + y = 5 → y = 2

Elimination Idea

Match coefficients (same or opposite), then subtract or add

If coefficients differ, multiply by suitable numbers

Substitution — Plug One Equation Into Another

📝 Substitution Example

①y = 2x - 1 …

①

②3x + y = 9 …

②

③Substitute

① into

②: 3x + (2x - 1) = 9

④5x - 1 = 9 → x = 2

⑤Then y = 2(2) - 1 = 3

Substitution Idea

Solve one equation for y → substitute into the other

Convenient if one variable already has coefficient 1

Number of Solutions

💡 How Many Solutions?

①Lines meet at one point → 1 solution (generic)

②Parallel lines → no solution (inconsistent)

③Same line → infinitely many solutions (dependent)

Work It Out

Example 1

Solve the system x + y = 5, x − y = 1 by elimination.

Adding the two equations eliminates y.

(x + y) + (x − y) = 5 + 1 ⇒ 2x = 6

Substitute x = 3 into one equation.

x = 3, y = 5 − 3 = 2

▸ x = 3, y = 2

Elimination matches one coefficient, then adds or subtracts to cancel a variable.

Example 2

Solve the system y = 2x − 1, 3x + y = 9 by substitution.

Substitute the first equation into y of the second.

3x + (2x − 1) = 9 ⇒ 5x = 10

Substitute x = 2.

x = 2, y = 2·2 − 1 = 3

▸ x = 2, y = 3

When one equation is y = (expression), substitution is convenient.

Exam Wrap-up

Solving Strategy

Elim: match coeffs → +/- Sub: y = ... → plug in

Both methods aim to eliminate one variable

Grade-8 school exam type

The system x + ay = 7, 2x − y = 4 has solution x = 3, y = 2. What is the constant a?

①1

②2

③3

④4

⑤5

▸ ② 2

Substitute the solution (3, 2) into the first equation.

3 + a·2 = 7

Solve for a.

2a = 4 ⇒ a = 2

🎯 Exam Points

①Elim vs Sub: if coeffs match easily → elim; if y = … already → sub

②When matching, watch signs (subtraction flips all)

③After solving, verify by plugging back into both equations

④Word problem: 2 unknowns → 2 conditions

⑤Graph: intersection coordinates = solution

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