Simultaneous Equations Revision Notes for VCE SSCE Mathematical Methods

Simultaneous Equations

What are simultaneous equations?

A linear equation with two unknowns, such as $2y + 3x = 10$ , does not have a single unique solution. Instead, this equation represents a relationship between pairs of numbers $(x, y)$ that satisfy the equation. When we plot all possible pairs of numbers that satisfy the equation on a graph, the result is a straight line. This is why we call it a linear relation.

When we have two linear equations and we graph them on the same set of axes, one of three things can happen. If the lines are not parallel, they will intersect at exactly one point. This intersection point represents the one unique pair of values $(x, y)$ that satisfies both equations at the same time - hence the term simultaneous equations.

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The intersection point can be found in three ways:

Graphically - by drawing both lines and identifying where they meet (though this method's accuracy depends on how carefully the graphs are drawn)
Algebraically - by solving the pair of equations using the substitution method or the elimination method (more accurate)

We will focus on the two algebraic techniques for solving simultaneous equations.

Method 1: Substitution method

The substitution method involves expressing one variable in terms of the other variable, then substituting this expression into the second equation. This reduces the problem to a single equation with one unknown, which can then be solved.

Steps for the substitution method

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Follow these steps to use the substitution method:

Choose one of the two equations and rearrange it to make one variable the subject
Substitute this expression into the other equation
Solve the resulting equation (which now contains only one variable)
Substitute the value you found back into one of the original equations to find the other variable
Check your answer by substituting both values into the equation you didn't use in step 4

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Worked Example: Solving Using Substitution

Solve the equations $2x - y = 4$ and $x + 2y = -3$

Solution:

Let's label the equations:

2x - y = 4 \quad \text{...(1)}

x + 2y = -3 \quad \text{...(2)}

From equation (2), we can rearrange to express $x$ in terms of $y$ :

x = -3 - 2y

Now substitute this expression for $x$ into equation (1):

2(-3 - 2y) - y = 4

Expand the brackets:

-6 - 4y - y = 4

Simplify:

-5y = 10

y = -2

Now substitute $y = -2$ into equation (2) to find $x$ :

x + 2(-2) = -3

x - 4 = -3

x = 1

Finally, check this solution using equation (1):

Left side: $2(1) - (-2) = 2 + 2 = 4$ ✓

Right side: $4$ ✓

The solution is x = 1 and y = -2, or we can write this as the point $(1, -2)$ .

Method 2: Elimination method

The elimination method involves adding or subtracting the two equations to eliminate one of the variables. This requires making the coefficients of one variable the same (or opposite) in both equations.

Steps for the elimination method

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Follow these steps to use the elimination method:

Choose which variable you want to eliminate
If necessary, multiply one or both equations by a number to make the coefficients of that variable equal or opposite
Add or subtract the equations to eliminate the chosen variable
Solve the resulting equation for the remaining variable
Substitute this value back into one of the original equations to find the other variable
Check your answer using the other equation

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Worked Example: Solving Using Elimination

Solve the equations $2x - y = 4$ and $x + 2y = -3$

Solution:

Let's label the equations:

2x - y = 4 \quad \text{...(1)}

x + 2y = -3 \quad \text{...(2)}

To eliminate $x$ , we need the coefficients of $x$ to be the same in both equations. Multiply equation (2) by 2:

2x - y = 4 \quad \text{...(1)}

2x + 4y = -6 \quad \text{...(2')}

Now subtract equation (2') from equation (1):

(2x - y) - (2x + 4y) = 4 - (-6)

2x - y - 2x - 4y = 4 + 6

-5y = 10

y = -2

Now substitute $y = -2$ into equation (2) to find $x$ :

x + 2(-2) = -3

x - 4 = -3

x = 1

The solution is x = 1 and y = -2, which is the same answer we obtained using the substitution method. This point $(1, -2)$ is where the two lines intersect on a graph.

The geometry of simultaneous equations

Two distinct straight lines can either be parallel to each other or meet at a single point. When we solve a system of two linear equations with two variables, there are three possible outcomes, each corresponding to a different geometric situation.

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Three Possible Cases for Simultaneous Equations:

Understanding the geometric interpretation helps explain why systems of equations can have different types of solutions.

Case 1: Unique solution (intersecting lines)

When two lines have different gradients, they intersect at exactly one point. This gives us a unique solution to the system of equations.

Example: $2x + y = 5$ and $x - y = 4$

Solution: $x = 3$ , $y = -1$

Geometry: Two lines meeting at a point

Case 2: No solutions (parallel lines)

When two lines have the same gradient but different $y$ -intercepts, they are parallel and never meet. This system has no solutions.

Example: $2x + y = 5$ and $2x + y = 7$

These equations cannot both be true at the same time.

Geometry: Distinct parallel lines

Case 3: Infinitely many solutions (coincident lines)

When two equations represent the exact same line (one equation is just a multiple of the other), every point on the line is a solution. This system has infinitely many solutions.

Example: $2x + y = 5$ and $4x + 2y = 10$

The second equation is just the first equation multiplied by 2, so they represent the same line.

Geometry: Two copies of the same line (coincident lines)

Word problems involving simultaneous equations

Many real-world problems involve finding two unknown quantities. We can solve these by setting up a system of simultaneous equations. The key is to carefully define your variables and translate the information in the problem into mathematical equations.

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Worked Example: Finding Two Numbers

The sum of two numbers is 24 and their difference is 96. Find the two numbers.

Solution:

Step 1: Define variables

Let $x$ and $y$ be the two numbers.

Step 2: Set up equations

The sum is 24:

x + y = 24 \quad \text{...(1)}

The difference is 96:

x - y = 96 \quad \text{...(2)}

Step 3: Solve using elimination

Add equations (1) and (2):

(x + y) + (x - y) = 24 + 96

2x = 120

x = 60

Step 4: Find the other variable

Substitute $x = 60$ into equation (1):

60 + y = 24

y = -36

Step 5: State the answer and check

The two numbers are 60 and -36.

Check in equation (2): $60 - (-36) = 60 + 36 = 96$ ✓

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Worked Example: Cost Problems

3 kg of jam and 2 kg of butter cost $29, and 6 kg of jam and 3 kg of butter cost $54. Find the cost per kilogram of jam and butter.

Solution:

Step 1: Define variables

Let $x$ = cost of 1 kg of jam (in dollars)

Let $y$ = cost of 1 kg of butter (in dollars)

Step 2: Set up equations

From the first purchase:

3x + 2y = 29 \quad \text{...(1)}

From the second purchase:

6x + 3y = 54 \quad \text{...(2)}

Step 3: Solve using elimination

Multiply equation (1) by 2:

6x + 4y = 58 \quad \text{...(1')}

Subtract equation (2) from equation (1'):

(6x + 4y) - (6x + 3y) = 58 - 54

y = 4

Step 4: Find the other variable

Substitute $y = 4$ into equation (2):

6x + 3(4) = 54

6x + 12 = 54

6x = 42

x = 7

Step 5: State the answer and check

Jam costs $7 per kg and butter costs $4 per kg.

Check with the original problem:

First purchase: $3 \times 7 + 2 \times 4 = 21 + 8 = 29$ ✓
Second purchase: $6 \times 7 + 3 \times 4 = 42 + 12 = 54$ ✓

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Key Points to Remember:

Simultaneous equations are pairs of equations with two unknowns that we solve together to find values that satisfy both equations
Substitution method: Rearrange one equation to express one variable in terms of the other, then substitute into the second equation
Elimination method: Multiply equations if needed to make coefficients equal, then add or subtract to eliminate one variable
Three possible outcomes: One solution (lines intersect), no solutions (lines are parallel), or infinitely many solutions (same line)
Word problems: Carefully define your variables, translate the problem into equations, solve, and always check your answer against the original problem

Simultaneous Equations (VCE SSCE Mathematical Methods): Revision Notes