Simultaneous Equations Revision Notes for Leaving Cert Mathematics

Simultaneous Equations

What are simultaneous equations?

Simultaneous equations are two or more equations that must be satisfied by the same values of the variables at the same time. When we have two linear equations with two unknowns (usually x and y), we need to find the values that make both equations true simultaneously.

For example, consider these equations:

$3x + y = 9$
$2x - y = 1$

Both equations are satisfied when $x = 2$ and $y = 3$ . We can verify this by substituting these values into both equations.

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The key to understanding simultaneous equations is that the same variable values must work for all equations in the system. If you have a solution where x = 2 and y = 3, these values must satisfy every equation in the system.

The elimination method

The elimination method is the primary technique for solving simultaneous equations. This method works by eliminating one variable so we can solve for the other.

Step-by-step process

Number the equations for convenience (label them ① and ②)
Make the coefficients equal for one variable by multiplying one or both equations
Add or subtract the equations to eliminate that variable
Solve the resulting equation for the remaining variable
Substitute this value back into either original equation to find the other variable
Verify your solution by checking both original equations

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The most critical step is making the coefficients equal for one variable. This often involves multiplying one or both equations by appropriate numbers. Choose the variable that requires the least manipulation to make your work easier.

Worked examples

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Worked Example 1: Basic elimination method

Solve: $2x - 5y = 9$ ... ① $3x + 2y = 4$ ... ②

Step 1: Make the y-coefficients equal by multiplying equation ① by 2 and equation ② by 5:

① × 2: $4x - 10y = 18$ ② × 5: $15x + 10y = 20$

Step 2: Add the equations to eliminate y: $4x - 10y + 15x + 10y = 18 + 20$ $19x = 38$ $x = 2$

Step 3: Substitute $x = 2$ into equation ①: $2(2) - 5y = 9$ $4 - 5y = 9$ $-5y = 5$ $y = -1$

Solution: $x = 2$ and $y = -1$

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Worked Example 2: Dealing with fractions

Solve: $3x - 2y = 19$ ... ① $\frac{x}{3} + \frac{y}{2} = 5$ ... ②

Step 1: Eliminate fractions in equation ② by multiplying by 6 (LCM of 3 and 2): $2x + 3y = 30$ ... ②

Step 2: Make x-coefficients equal by multiplying ① by 2 and ② by -3: ① × 2: $6x - 4y = 38$ ② × -3: $-6x - 9y = -90$

Step 3: Add to eliminate x: $-13y = -52$ $y = 4$

Step 4: Substitute $y = 4$ into equation ①: $3x - 8 = 19$ $3x = 27$ $x = 9$

Solution: $x = 9$ and $y = 4$

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Worked Example 3: Real-world application with geometry

For an equilateral triangle, all sides must be equal in length. If the sides are:

Side AB: $3x + 2y$
Side AC: $3y - 1$
Side BC: $11 + x - 2y$

We can form simultaneous equations by setting any two sides equal: $3x + 2y = 3y - 1$ ... ① $3x + 2y = 11 + x - 2y$ ... ②

Solving these equations will give us the values of x and y, allowing us to find the actual length of each side.

Common exam tips

Mastering simultaneous equations requires both understanding the method and avoiding common pitfalls. Here are the most important strategies for exam success:

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Essential Exam Strategies:

Always verify your answer by substituting the values back into both original equations
Clear fractions first by multiplying through by the LCM of the denominators
Keep your work organised by numbering equations and showing each step clearly
Check for simple arithmetic errors - these are common mistakes in exams
Look for the easiest elimination - choose the variable that requires the least manipulation

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Key Points to Remember:

Simultaneous equations are equations that share the same solution values for their variables
The elimination method works by making coefficients equal and then adding/subtracting to eliminate one variable
Always verify your solution by substituting back into both original equations
Clear fractions first to make calculations easier and reduce errors
Real-world applications often involve setting expressions equal based on given conditions (like equal sides of an equilateral triangle)

Simultaneous Equations (Leaving Cert Mathematics): Revision Notes