System of Linear Equations

A system of equations hunts for the values that satisfy two equations in two unknowns at once. Buy apples and pears, then write one equation for the count and one for the cost to find how many of each. You can substitute one letter into the other equation, or add and subtract to cancel a letter. Here the point where two line graphs cross is exactly the solution.

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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Inequalities

Linear Function Graph

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