Simultaneous Equations Revision Notes for Grade 11 NSC Matric Mathematics

Simultaneous Equations

What are simultaneous equations?

Simultaneous equations are a system of two or more equations that must be solved together to find the values of unknown variables. These equations share the same variables and must all be satisfied at the same time.

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To solve for the values of n unknown variables, you need a system of n independent equations. For example, to find two unknown variables (x and y), you need two independent equations.

An example of a system with one linear and one quadratic equation is:

$y - 2x = -4$
$x^2 + y = 4$

Three methods for solving simultaneous equations

There are three main approaches to solving simultaneous equations:

Substitution method - Express one variable in terms of another and substitute
Elimination method - Make coefficients equal and eliminate one variable
Graphical method - Plot both equations and find intersection points

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The graphical method is particularly useful for showing where two equations intersect visually, making it easier to understand the geometric relationship between the equations.

Method 1: Solving by substitution

The substitution method works by reducing the system to one equation with one unknown variable. Here's the step-by-step process:

Choose the simpler equation and express one variable in terms of the other
Substitute this expression into the second equation
Solve the resulting equation which now has only one variable
Substitute back to find the value of the other variable
Check your answer by verifying both original equations are satisfied

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Worked Example: Substitution Method

Question: Solve for x and y:

$y - 2x = -4$ ... (1)
$x^2 + y = 4$ ... (2)

Solution:

Step 1: Make y the subject of equation (1)

$y = 2x - 4$

Step 2: Substitute into equation (2)

$x^2 + (2x - 4) = 4$
$x^2 + 2x - 8 = 0$

Step 3: Factorise and solve

$(x + 4)(x - 2) = 0$
Therefore $x = -4$ or $x = 2$

Step 4: Find corresponding y-values

If $x = -4$ :

$y = 2(-4) - 4 = -12$

If $x = 2$ :

$y = 2(2) - 4 = 0$

Step 5: Check and write final answer

The solutions are $x = -4, y = -12$ and $x = 2, y = 0$ . These give the coordinate pairs $(-4, -12)$ and $(2, 0)$ where the equations intersect.

Method 2: Solving by elimination

The elimination method works by making the coefficients of one variable equal in both equations, then adding or subtracting to eliminate that variable.

Steps for elimination:

Make coefficients equal for one of the variables in both equations
Add or subtract the equations to eliminate that variable
Solve the resulting equation with one unknown
Substitute back to find the other variable value
Check that both points satisfy the original equations

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Worked Example: Elimination Method

Question: Solve for x and y:

$y = x^2 - 6x$ ... (1)
$y + \frac{1}{2}x - 3 = 0$ ... (2)

Solution:

Step 1: Make y the subject of equation (2)

$y = -\frac{1}{2}x + 3$

Step 2: Set the equations equal and solve for x

$x^2 - 6x = -\frac{1}{2}x + 3$
$x^2 - 6x + \frac{1}{2}x - 3 = 0$
$2x^2 - 12x + x - 6 = 0$ (multiply by 2)
$2x^2 - 11x - 6 = 0$
$(2x + 1)(x - 6) = 0$

Therefore $x = -\frac{1}{2}$ or $x = 6$

Step 3: Find corresponding y-values

If $x = -\frac{1}{2}$ :

$y = -\frac{1}{2}(-\frac{1}{2}) + 3 = \frac{1}{4} + 3 = 3\frac{1}{4}$

If $x = 6$ :

$y = -\frac{1}{2}(6) + 3 = 0$

Final answer: The solutions are $(-\frac{1}{2}, 3\frac{1}{4})$ and $(6, 0)$ .

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Worked Example: Another Elimination Example

Question: Solve for x and y:

$y = \frac{5}{x-2}$ ... (1)
$y + 1 = 2x$ ... (2)

Solution:

Step 1: Rearrange equation (2)

$y = 2x - 1$

Step 2: Set equations equal

$\frac{5}{x-2} = 2x - 1$
$(2x - 1)(x - 2) = 5$
$2x^2 - 5x + 2 = 5$
$2x^2 - 5x - 3 = 0$
$(2x + 1)(x - 3) = 0$

Therefore $x = -\frac{1}{2}$ or $x = 3$

Step 3: Find y-values

For $x = -\frac{1}{2}$ : $y = 2(-\frac{1}{2}) - 1 = -2$
For $x = 3$ : $y = 2(3) - 1 = 5$

Final answer: The solutions are $(-\frac{1}{2}, -2)$ and $(3, 5)$ .

Method 3: Solving graphically

The graphical method involves plotting both equations on the same coordinate system. The solutions are the coordinates where the graphs intersect.

Steps for graphical method:

Rearrange both equations to make y the subject
Plot both graphs on the same coordinate system
Identify intersection points from the graph
Read off coordinates of intersection points
Check that these points satisfy both original equations

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Worked Example: Graphical Method

Question: Solve graphically for x and y:

$y + x^2 = 1$ ... (1)
$y - x + 5 = 0$ ... (2)

Solution:

Step 1: Make y the subject of both equations

From equation (1): $y = -x^2 + 1$
From equation (2): $y = x - 5$

Step 2: Plot both graphs on the same axes

The first equation is a downward-opening parabola with vertex at (0, 1).
The second equation is a straight line with slope 1 and y-intercept -5.

Step 3: Find intersection points

From the graph, the intersection points are at $(-3, -8)$ and $(2, -3)$ .

Exam tips and common mistakes

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Key exam strategies:

Always check your solutions by substituting back into both original equations
When using substitution, choose the simpler equation to rearrange first
For elimination, look for coefficients that are easy to make equal
In graphical methods, draw accurate graphs and clearly mark intersection points

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Common exam traps:

Forgetting to find both x and y values for each solution
Making arithmetic errors when expanding brackets
Not checking that solutions satisfy both original equations
Mixing up which method is most appropriate for the given equations

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When to use each method:

Substitution: When one equation is easily rearranged (especially linear equations)
Elimination: When coefficients can be easily matched
Graphical: When asked specifically, or to visualise the relationship between equations

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Key Points to Remember:

Simultaneous equations require finding values that satisfy all equations in the system at once
Three main methods: substitution (rearrange and substitute), elimination (make coefficients equal), and graphical (plot and find intersections)
Always check your answers by substituting back into the original equations
Choose the most efficient method based on the form of the equations given
Graphical solutions show the geometric relationship between the equations and their intersection points

Simultaneous Equations (Grade 11 NSC Matric Mathematics): Revision Notes