Simultaneous Linear Equations
ICSE Class 9 · Mathematics · Chapter 6
What this chapter covers
- Linear equations in two variables
- Simultaneous linear equations and their solution
- Solving by the method of elimination
- Solving by the method of substitution
- Solving by the method of cross-multiplication
- Rearranging equations before solving
- Framing equations from word problems
- Problems on ages, digits and fractions
- Problems on speed and on measurement
- Common mistakes and how to avoid them
Key definitions
Simultaneous linear equations are two or more linear equations in the same variables that must be satisfied by the same values of those variables at the same time.
Key points
- A linear equation in two variables has the form ax + by + c = 0 with a and b not both zero; it has infinitely many solutions.
- Simultaneous linear equations must be satisfied by the same values of the variables at the same time; their common solution is an ordered pair (x, y).
- Every algebraic method removes one variable, solves for the other, and then substitutes back.
- Elimination: make one variable's coefficients equal in size, then add (opposite signs) or subtract (same signs).
- For interchanged coefficients, add and subtract the equations to get x + y and x - y.
- Substitution: make one variable the subject of one equation and put it, in brackets, into the other.
- Cross-multiplication: for a1x + b1y + c1 = 0 and a2x + b2y + c2 = 0, (x)/(b1c2 - b2c1) = (y)/(c1a2 - c2a1) = (1)/(a1b2 - a2b1), provided a1b2 - a2b1 ≠ 0.
- Clear fractions, decimals and brackets first, so that every equation is in the form ax + by = c.
- In word problems, name both unknowns with units, turn each fact into one equation, and check the answer against the words.
- A two-digit number with digits x and y is 10x + y; ages change by the same number of years for everyone; downstream speed is x + y and upstream speed is x - y.
- Always check the solution in both original equations.
Questions answered
Define simultaneous linear equations.
Simultaneous linear equations are two or more linear equations in the same variables that must be satisfied by the same values of the variables at the same time.
Is (2, -1) a solution of the pair 3x + 4y = 2 and x - 2y = 4? Give a reason.
Yes. 3(2) + 4(-1) = 6 - 4 = 2 and 2 - 2(-1) = 2 + 2 = 4, so the pair satisfies both equations.
Why does a single linear equation in two variables not fix the values of the variables?
For any value chosen for one variable, the equation gives a matching value of the other, so it has infinitely many solutions. A second, independent equation is needed to pick out one pair.
Solve by elimination: 5x + 3y = 19 and 2x - 3y = -5.
Adding, 7x = 14, so x = 2. Then 10 + 3y = 19, so y = 3. Check: 4 - 9 = -5. Solution x = 2, y = 3.
Solve by substitution: x = 3y + 1 and 2x + 5y = 24.
2(3y + 1) + 5y = 24, so 11y + 2 = 24 and y = 2. Then x = 7. Check: 14 + 10 = 24. Solution x = 7, y = 2.
Solve by cross-multiplication: 2x - y - 3 = 0 and x + 3y - 5 = 0.
a1 = 2, b1 = -1, c1 = -3; a2 = 1, b2 = 3, c2 = -5. The denominators are b1c2 - b2c1 = 5 + 9 = 14, c1a2 - c2a1 = -3 + 10 = 7 and a1b2 - a2b1 = 6 + 1 = 7. So x = (14)/(7) = 2 and y = (7)/(7) = 1.
Solve 23x + 17y = 63 and 17x + 23y = 57.
Adding, 40x + 40y = 120, so x + y = 3. Subtracting, 6x - 6y = 6, so x - y = 1. Hence x = 2, y = 1.
Solve (x)/(4) + (y)/(2) = 3 and (x)/(2) - (y)/(4) = 1.
Multiplying each by 4 gives x + 2y = 12 and 2x - y = 4. Then y = 2x - 4, so x + 4x - 8 = 12, giving x = 4 and y = 4.
Past board exam questions
ICSE 2019 sample paper · 3 marks
Solve following pairs of linear equations using cross-multiplication method: 5x-3y=2 4x+7y=-3
Show answer
Write as 5x-3y-2=0 and 4x+7y+3=0.
(x)/((-3)(3)-(7)(-2)) = (y)/((-2)(4)-(3)(5)) = (1)/((5)(7)-(4)(-3))
(x)/(5) = (y)/(-23) = (1)/(47)
x = (5)/(47), y = -(23)/(47)
ICSE 2019 sample paper · 3 marks
Mr. Mohan has ₹ 256 in the form of ₹1 and ₹ 2 coins. If the number of ₹ 2 coins are three more than twice the number of ₹ 1 coins, find the total value of ₹ 2 coins.
Show answer
Let the number of ₹1 coins be x and the number of ₹2 coins be y.
x + 2y = 256 and y = 2x + 3.
x + 2(2x + 3) = 256 ⇒ 5x + 6 = 256 ⇒ x = 50, y = 103.
Total value of ₹2 coins = 103 × 2 = ₹206.