Simultaneous (Linear) Equations

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A linear equation in two variables is an equation of the form ax + by + c = 0, where: - a, b, and c are constants (real numbers) - x and y are variables - Each variable has degree 1 (power 1) Example:

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Simultaneous linear equations are two or more linear equations that contain the same variables and are solved together to find common values of the variables. Example: 2x + 3y = 7 4x - y = 1 These e

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To verify: x = 3, y = 5 Checking in first equation: 2(3) - 5 = 6 - 5 = 1 ✓ Checking in second equation: 3(3) + 5 = 9 + 5 = 14 ✓ Therefore, x = 3 and y = 5 is the solution as it satisfies both equat

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The three algebraic methods are: 1. Method of elimination by substitution 2. Method of elimination by equating coefficients 3. Method of cross-multiplication Each method is useful in different situa

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Steps for substitution method: Step 1: From either equation, express one variable in terms of the other Step 2: Substitute this expression in the other equation and solve for one variable Step 3: Sub

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Solution: Step 1: From x + y = 7, we get y = 7 - x Step 2: Substitute in second equation: 3x - 2(7 - x) = 11 3x - 14 + 2x = 11 5x = 25 x = 5 Step 3: y = 7 - x = 7 - 5 = 2 Answer: x = 5, y = 2

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Steps for elimination method: Step 1: Multiply one or both equations by suitable numbers to make coefficients of one variable numerically equal Step 2: Add or subtract the equations to eliminate one

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Solution: Step 1: Multiply first equation by 5 and second by 3: 15x - 20y = 50 15x - 9y = 72 Step 2: Subtract first from second: -9y - (-20y) = 72 - 50 11y = 22 y = 2 Step 3: Substitute y = 2 in fir