Linear Equations • Topic 3 of 3

Word Problems with Linear Equations

What are Simultaneous Equations? Simultaneous equations are two or more equations that share the same variables. They are solved together (simultaneously) to find values that satisfy all equations at the same time.

Example: $x + y = 10$ $x - y = 4$

Why do we need them? Sometimes one equation with two variables has infinitely many solutions. Adding another equation gives a unique solution.

Methods to Solve Simultaneous Equations:

Method	How it works	Best for
Elimination Method	Add/subtract equations to eliminate one variable	When coefficients are friendly
Substitution Method	Solve one equation for one variable, substitute into other	When one variable is easily isolated

Elimination Method Steps:

Align equations so like terms are stacked
Multiply one or both equations to make coefficients of one variable opposites
Add or subtract to eliminate that variable
Solve for the remaining variable
Substitute back to find the other variable

Substitution Method Steps:

Solve one equation for one variable (e.g., $x = something$)
Substitute this expression into the other equation
Solve for the remaining variable
Substitute back to find the other variable

Solved Examples

Worked Example

Solve: $2x + y = 7$ and $x - y = 2$

Solution- Align equations: $2x + y = 7$ $x - y = 2$ - Add the equations: $(2x + x) + (y - y) = 7 + 2$ - $3x = 9 \implies x = 3$ - Substitute $x = 3$ into $x - y = 2$: $3 - y = 2$ - $-y = -1 \implies y = 1$ - Check in first: $2(3) + 1 = 6 + 1 = 7$ ✓ - **Answer:** $x = 3$, $y = 1$ *Example 2 (Substitution Method): Solve: $x = 2y + 3$ and $3x - 2y = 5$ Solution: - First equation gives $x = 2y + 3$ - Substitute into second: $3(2y + 3) - 2y = 5$ - $6y + 9 - 2y = 5$ - $4y + 9 = 5$ - $4y = -4 \implies y = -1$ - Substitute back: $x = 2(-1) + 3 = -2 + 3 = 1$ - Check: $3(1) - 2(-1) = 3 + 2 = 5$ ✓ - **Answer:** $x = 1$, $y = -1$ *Example 3 (Real-Life Application): The sum of two numbers is 25 and their difference is 7. Find the numbers. Solution: - Let the numbers be $x$ and $y$ (with $x > y$) - Equation 1: $x + y = 25$ - Equation 2: $x - y = 7$ - Add equations: $(x+y) + (x-y) = 25 + 7$ - $2x = 32 \implies x = 16$ - Substitute: $16 + y = 25 \implies y = 9$ - Check: $16 - 9 = 7$ ✓ - **Answer:** The numbers are 16 and 9

Key Points

Simultaneous equations share variables and are solved together
Elimination method: add/subtract equations to remove one variable
Substitution method: solve for one variable, substitute into other
Both methods give the same solution; choose based on convenience
Solution is an ordered pair $(x, y)$ that satisfies both equations
Graphically, the solution is the intersection point of two lines
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Quick Check — 4 MCQs

Tap an option to check your answer0 / 4

Q1."$5$ more than a number $x$" is written as:

Explanation: $x+5$.

Q2."Twice a number $x$" is:

Explanation: $2x$.

Q3.The sum of a number and $7$ is $15$. The number is:

Explanation: $x=8$.

Q4.The first step in a word problem is to:

Explanation: Define a variable.

Practice more MCQs → 📝 Assignment · 35 marks