Introduction
Simultaneous means to occur, happen, or be done at the same time.
Simultaneous equations are a set of different equations that are all true when the variables have the same value in each equation. A set of simultaneous equations is often called a system of equations.
To solve a set of simultaneous equations, you need to find the values of the variables that satisfy all the equations simultaneously. It is not always possible to solve equations simultaneously.
Simultaneous equations are used to solve everyday problems that involve more than one variable. The following are some common situations in which you could use simultaneous equations to find the solution to the problem.
https://sciencing.com/everyday-examples-situations-apply-quadratic-equations-10200.html
Speed, distance and time
You can use simultaneous equations to work out the best paths for your running training by considering how far you run and how fast you go at different parts of the route. You can set different targets using these equations, such as increasing running time to improve your endurance or maximising speed for better performance.
Simultaneous equations can also help you find and compare the speed, distance, and time when travelling by car, plane or train. For instance, an air traffic controller uses simultaneous equations to make sure that planes don't crash when they come in to land at the same time.
Comparing product and service costs
The best phone plan
The best care rental
The best loan
Understanding economics
Demand and supply
Loans
Investments
Job benefits
Overview
In this topic, you will work with linear equations. In other words, all the variables are to the power of 1.
The following are examples of linear equations:
$$\begin{aligned}2x+5y&=10\\x-8y+z&=3\end{aligned}$$
Each term in a linear equation consists of either a constant (from the previous examples, 10 and 3) or a multiple of a variable to power of 1 (x, y and z). Variable terms include at least one pronumeral and the coefficient. Here, the x-coefficients are 2 and 1; the y-coefficients are 5 and −8; and the z-coefficient is 1.
Variables are sometimes called unknowns since their values are not (yet) known.
The linear equation \(2x+5y=10\) has two unknown variables, x and y.
The linear equation \(x-8y+z=3\) has three unknown variables, x, y and z.
Check your understanding
Complete the following three (3) tasks. Click the arrows to navigate between the tasks. You may attempt this activity as many times as you like.
All tasks refer to the following equation
$$−7p+q-4r=6$$
Learning Aims – Simultaneous Equations
By the end of this topic, you should be able to:
- Solve 2 × 2 system of equations
- Solve 3 × 3 system of equations
- Understand and describe geometric interpretations
- Use simultaneous equations to solve problems
A linear equation with two unknown variables represents a straight line in 2-dimensional space (in other words, in the plane). This is why these types of equations are called linear equations.
For example, the following linear equation can be rearranged into the familiar straight line formula.
$$\begin{aligned}2x+5y&=10\\y&=\frac{-2}{5}\times2\end{aligned}$$
Where the gradient and y-intercept can be read directly from the equation.
$$\begin{aligned}\mathsf{Gradient}&=\frac{-2}{5}\\y-\mathsf{intercept}&= 2\end{aligned}$$
When solving two linear equations with two unknown variables, your aim is to find a single value for each variable that satisfies each of the simultaneous equations.
Consider the two simultaneous linear equations with two unknown variables:
$$\begin{aligned}2x+5y&=17\\5x-y&=29\end{aligned}$$
When solved, the following values for each variable are obtained.
$$\begin{aligned}x=6, y=1\end{aligned}$$
These values satisfy the pair of simultaneous equations because the same values make both equations true.
Geometric Interpretations - Two Linear Equations with Two Unknowns
There are three possible outcomes when attempting to solve two linear equations with two unknown variables simultaneously:
- A unique solution
- Multiple solutions
- No solution.
These three outcomes are summarised in the following table.
| Outcome | Unique Solution | Multiple Solutions | No Solution |
|---|---|---|---|
| Geometric Interpretation | Non-parallel straight lines | Coincident straight lines | Parallel straight lines |
| Graphic Representation |  |
 |
 |
| Equations | $$\begin{aligned}x+y&=2\\10x-y&=-13\end{aligned}$$ | $$\begin{aligned}x+y&=2\\3x+3y&=6\end{aligned}$$ | $$\begin{aligned}x+y&=2\\2x+2y&=10\end{aligned}$$ |
| Solution | \(x=-1,\;y=3\) | Reduces to \(0=0\) | Reduces to \(0=±6\) |
| Description | Two lines intersect at (-1, 3) | Dependent equations | A contradiction |
| Interpretation | Equations are consistent | Equations are consistent | Equations are inconsistent |
Methods for solving a 2×2 system of equations
The two main methods for solving simultaneous equations are by:
- substitution
- elimination.
Substitution method
The substitution method uses one equation to substitute (replace) one of the variables in the other equation.
Example 1
Solve the following equations simultaneously.
$$\begin{aligned}y&=3x-1 &\mathsf{Equation} (1)\\x+2y&=12 &\mathsf{Equation} (2)\end{aligned}$$
| Step instrucions | Calculations |
|---|---|
|
1. Substitute equation (1) into equation (2). 2. Expand and collect like terms. 3. Solve. |
$$\begin{aligned}x+2(3x-1)&=12\\7x&=14\\x&=2\end{aligned}$$ |
| 4. Substitute the x-value into either equation and solve for y. | $$\begin{aligned}y&=3(2)-1\\&=5\end{aligned}$$ |
Hence, a unique solution \(x=2\) and \(y=5\) is obtained.
Elimination method
The elimination method seeks to eliminate (remove) the term containing the same variable from each equation.
Example 2
Solve the following equations simultaneously.
$$\begin{aligned}3x-5y&=2 &\mathsf{Equation} (1)\\2x-3y&=4 &\mathsf{Equation} (2)\end{aligned}$$
| Step instrucions | Calculations |
|---|---|
| 1. First, decide which variable to eliminate. | In this example, the x-variable will be eliminated. To do this, multiply the equations by a constant to make the coefficients of the x variable the same in both equations. |
| 2. Multiply equation (1) by 2. | $$\begin{aligned}3x-5y&=2 &\mathsf{Equation} (1)\times2\\6x-10y&=4 &\mathsf{Equation} (3)\end{aligned}$$ |
| 3. Multiply equation (2) by 3. | $$\begin{aligned}2x-3y&=4 &\mathsf{Equation} (2)\times3\\6x-9y&=12 &\mathsf{Equation} (4)\end{aligned}$$ |
| 4. Subtract equation (4) from equation (3) and solve for y. | $$\begin{aligned}2x-3y&=4 &\mathsf{Equation} (3)\\- 6x-9y&=12 &\mathsf{Equation} (4)\\-y&=-8\\y&=8\end{aligned}$$ |
| 5. Substitute the y-value into either equation and solve for x. | $$\begin{aligned}6x-10(8)&=4\\6x&=4+80\\x&=\frac{84}{6}\\&=14\end{aligned}$$ |
Hence, a unique solution \(x=14\) and \(y=8\) is obtained.
Exercises
Exercise 1: 2 × 2 system of equations (unique solution)
Solve the following simultaneous equations:
- \(\begin{aligned}2x+3y&=1\\3x-y&=7\\\end{aligned}\)
- \(\begin{aligned}x-y&=8\\5x+2y&=61\\\end{aligned}\)
- \(\begin{aligned}4x+3y+14&=0\\-5x+2y-29&=0\\\end{aligned}\)
Exercise 2: 2 × 2 system of equations
Solve the following simultaneous equations. Give the unique solution, if it exists. Otherwise, state whether the equations are dependent or inconsistent.
- \(\begin{aligned}x-4y&=-5\\2x+3y&=23\\\end{aligned}\)
- \(\begin{aligned}x-y&=8\\5x-5y&=40\\\end{aligned}\)
- \(\begin{aligned}4x+2y&=24\\-3x-1.5y&=30\\\end{aligned}\)
- \(\begin{aligned}2x+3y&=1\\6x+9y&=3\\\end{aligned}\)
- \(\begin{aligned}x-7y&=8\\5x-35y&=61\\\end{aligned}\)
- \(\begin{aligned}4x+2y&=22\\-3x-5y&=-27\\\end{aligned}\)
Select the following heading to view the answers to the previous Exercises.
Exercise 1
- \(x=2,\;y=-1\)
- \(x=11,\;y=3\)
- \(x=-5,\;y=2\)
Exercise 2
- Unique solution: \(x=7,\;y=3\)
- Infinite number of solutions: Dependent equations (coincident lines)
- No solution: Inconsistent equations (parallel lines)
- Infinite number of solutions: Dependent equations (coincident lines)
- No solution: Inconsistent equations (parallel lines)
- Unique solution: \(x=4,\;y=3\)
A linear equation with three unknown variables represents a plane in 3-dimensional space (3D for short).
For example, the linear equation \(x+3y+z=3\) represents the plane on the xyz-axes shown in the following diagram.
Three linear equations are required to solve equations with three unknown variables.
As before, the aim is to obtain a single value for each variable to satisfy each equation simultaneously.
Three simultaneous equations with three unknown variables:
$$\begin{aligned}x+3y-2z&=-8\\2x-y+3z&=19\\5x+2y-z&=9\end{aligned}$$
When solved, \(x=3\), \(y=-1\) and \(z=4\) are obtained.
These values are said to satisfy the three simultaneous equations because they are true for all three equations simultaneously.
Geometric Interpretations - Three Linear Equations with Three Unknowns
As with simultaneously solving two linear equations with two unknowns, there are three possible outcomes when attempting to simultaneously solve three linear equations with three unknown variables:
- A unique solution
- Multiple solutions
- No solution.
1. A unique solution
There is one set of values for the unknowns that makes all three equations true because there is a single point where all three planes intersect, as shown in the following diagram.
For example, consider the following simultaneous equations.
$$\begin{aligned}x + 3y - 2z &= -8 \\2x - y + 3z &= 19 \\5x + 2y - z &= 9\end{aligned}$$
Clue: There should be no pattern in the coefficients and constants.
Geometric interpretation: Equations are consistent.
2. Multiple solutions
There are two ways a set of three simultaneous linear equations can have multiple solutions. Either all three planes are identical, or all three planes intersect on the same line.
Three coincident planes
When all three planes are coincident, the planes are identical and completely overlap, as shown in the following diagram.
For example, consider the following simultaneous equations.
$$\begin{aligned}x-y+z&=3\\2x-2y+2z&=6\\3x-3y+3z&=9\end{aligned}$$
Clue: There is a linear pattern in the coefficients and constants. In this example, equations (2) and (3) are multiples of equation (1).
Geometric interpretation: Equations are consistent.
Three planes intersecting at a common line
When planes are rotated around the same axis, they intersect along that axis - a common line, as shown in the following diagram.
For example, consider the following simultaneous equations.
$$\begin{aligned}2x+3y-2z&=5\\-x+y+z&=4\\x+4y-z&=9\end{aligned}$$
Clue: There is a linear pattern in the coefficients and constants.
Geometric interpretation: Equations are consistent.
No solution
There are three cases in which three simultaneous linear equations have an outcome of no solutions:
- None of the planes intersect at any point
- Two planes intersect with the third but not with each other
- Each plane intersects at unique points with the others - three separate points of intersection.
Three parallel planes
When all three planes are parallel, they cannot overlap at any point, as shown in the following diagram.
For example, consider the following simultaneous equations.
$$\begin{aligned}x+y-2z&=1\\x+y-2&=4\\3x+3y-6z&=2\end{aligned}$$
Clues: For all three equations:
- Coefficients of corresponding variables in an equation are a multiple of these in another equation.
- Constant terms do not follow the same pattern.
Geometric interpretation: Equations are inconsistent.
Two parallel planes with one intersecting plane
When two planes are parallel but are both intersected by the third, there is no point at which all three planes can intersect, as shown in the following diagram.
For example, consider the following simultaneous equations.
$$\begin{aligned}2x-y+5z&=3\\2x-y+5z&=7\\3x+2y-z&=12\end{aligned}$$
Clues: One equation is distinct, while for the other two equations:
- Coefficients of corresponding variables in an equation are a multiple of these in another equation.
- Constant terms do not follow the same pattern.
Geometric interpretation: Equations are inconsistent.
Each plane is parallel to the intersection of the other two planes
When each plane intersects with the other two, but on a different line, the planes form a triangle and cannot share one single intersection point, as shown in the following diagram.
For example, consider the following simultaneous equations.
$$\begin{aligned}2x-3y-3z&=1\\x+y+z&=-6\\3x-2y-2z&=4\end{aligned}$$
Clues: One equation is distinct, while for the other two equations:
- The coefficients of a variable have a simple linear rule.
- Constant terms do not follow the same pattern.
Geometric interpretation: Equations are inconsistent.
Methods for solving a 3×4 system of equations
You can use elimination or substitution methods to simultaneously solve three linear equations with three unknowns.
The following examples demonstrate using both methods to solve the same system of equations.
Example 3 - Substitution method
This method substitutes (replaces) one variable in two of the equations with the third equation.
For example, solve the following equations simultaneously.
$$\begin{aligned}x+3y-2z&=-8\;&(1)\\2x-y+3z&=19\;&(2)\\5x+2y-z&=9\;&(3)\end{aligned}$$
| Step Instructions | Calculations |
|---|---|
| 1. Rearrange equation (1) to make x the subject. | $$\begin{aligned}x+3y-2z&=-8\;&(1)\\x&=-3y+2z-8\;&(4)\end{aligned}$$ |
|
2. Substitute equation (4) into equations (2) and (3). 3. Expand and collect like terms. 4. Simplify. Note: Equations (5) and (6) now form a system of equations of two linear equations with two unknowns. |
$$\begin{aligned}2(-3y+2z-8)-y+3z&=19\;&(2)\\-6y+4z-16-y+3z&=19\\-7y+7z&=35\\y-z&=-5\;&(5)\\\\5(-3y+2z-8)+2y-z&=9\;&(3)\\-15y+10z-40+2y-z&=9\\-7y+7z&=35\\13y-9z&=-49\;&(6)\end{aligned}$$ |
|
5. Solve equations (5) and (6) using the substitution or elimination method. Using substitution:
|
$$\begin{aligned}y-z&=-5\;&(5)\\y&=z-5\\13(z-5)-9z&=-49\;&(6)\\z&=4\\\\y-(4)&=-5\;&(5)\\y&=-1\\\\x+3(-1)-2(4)&=-8\;&(1)\\x-11&=-8\\x&=3\end{aligned}$$ |
The unique solution for this system of equations is \(x=3,\;y=-1\;z=4\).
Example 4 - Elimination method
This method aims to eliminate (remove) a term from each equation containing the same variable.
For example, solve the following equations simultaneously.
$$\begin{aligned}x+3y-2z&=-8\;&(1)\\2x-y+3z&=19\;&(2)\\5x+2y-z&=9\;&(3)\end{aligned}$$
| Step Instructions | Calculations |
|---|---|
|
1. Decide which variable to eliminate. in this example, x. 2. Multiply equation (1) by 2. |
$$\begin{aligned}x+3y-2z&=-8\;&(1)\times2\\2x+6y-4z&=-16\;&(4)\end{aligned}$$ |
| 2. Subtract equation (2) from equation (4). | $$\begin{aligned}2x+6y-4z&=-16\;&(4)\\- 2x-y+3z&=19\;&(2)\\7y-7z&=-35\\y-z&=-5\;&(5)\end{aligned}$$ |
| 3. Multiply equation (1) by 5. | $$\begin{aligned}x+3y-2z&=-8\;&(1)\times5\\5x+15y-10z&=-40\;&(6)\end{aligned}$$ |
|
4. Subtract equation (3) from equation (6). Note: Equations (5) and (7) now form a system of equations of two linear equations with two unknowns. |
$$\begin{aligned}5x+15y-10z&=-40\;&(6)\\- 5x+2y-z&=9\;&(3)\\13y-9z&=-49\;&(7)\end{aligned}$$ |
|
5. Solve equations (5) and (7) using the substitution or elimination method. Using elimination:
|
$$\begin{aligned}y-z&=-5\;&(5)\times13\\13y-13z&=-65\;&(8)\\- 13y-9z&=-49\;&(7)\\z&=4\\\\y-(4)&=-5\\y&=-1\;&(5)\\\\2x-(-1)+3(4)&=19\;&(2)\\2x+1+12&=19\\x=3\end{aligned}$$ |
The unique solution for this system of equations is still \(x=3,\;y=-1\;z=4\).
Exercises
Exercise 3: 3 × 3 system of equations (unique solution)
Solve the following simultaneous equations:
- \(\begin{aligned}x+y+z&=6\\7x+y-z&=8\\2x-3y+5z&=3\end{aligned}\)
- \(\begin{aligned}2x-2y+z&=13\\3x+y+3z&=17\\2x+3y-2z&=5\\\end{aligned}\)
- \(\begin{aligned}x-y-z&=2\\2x+2y+3z&=26\\3x-y+4z&=26\\\end{aligned}\)
- \(\begin{aligned}-2x+y+z&=-5\\4x-y+3z&=10\\2x-2y+2z&=13\\\end{aligned}\)
- \(\begin{aligned}3x-3y+z&=15\\x+y+z&=5\\5x+8y-3z&=52\\\end{aligned}\)
- \(\begin{aligned}3x+5y+3z&=2\\-x-y+5z&=10\\2x+y+z&=4\\\end{aligned}\)
Exercise 4: 3 × 3 system of equations
Solve the following simultaneous equations. Give the unique solution, if it exists. Otherwise, state whether the equations are dependent or inconsistent. For each system of equations, provide a geometric interpretation.
- \(\begin{aligned}x+y+z&=6\\2x+3y+z&=11\\3x+2y+2z&=13\\\end{aligned}\)
- \(\begin{aligned}x+4y-z&=3\\-2x-8y+2z&=-6\\3x+12y-3z&=9\\\end{aligned}\)
- \(\begin{aligned}3x-6y-18z&=6\\x-2y-6z&=-2\\2x-4y-12z&=14\\\end{aligned}\)
- \(\begin{aligned}3x-2y-2z&=4\\7x+y-z&=8\\10x-y-3z&=10\\\end{aligned}\)
- \(\begin{aligned}2x+3y-5z&=0\\3x+2y-4z&=-2\\4x+y-z&=2\\\end{aligned}\)
- \(\begin{aligned}x-5y-3z&=23\\2x+2y+3z&=26\\3x-15y-9z&=65\\\end{aligned}\)
Select the following heading to view the answers to the previous Exercises.
Exercise 3
- \(x=1,\;y=3,\;z=2\)
- \(x=5,\;y=-1,\;z=1\)
- \(x=7,\;y=3,\;z=2\)
- \(x=0.5,\;y=-5,\;z=1\)
- \(x=7,\;y=1,\;z=-3\)
- \(x=2,\;y=-2,\;z=2\)
Exercise 4
- Unique solution: \(x=1,\;y=2,\;z=3\)
- Multiple solutions: Dependent equations forming three coincident planes.
- No solution: Inconsistent equations forming three parallel planes.
- No solution: Inconsistent equations forming three plane, where each plane is parallel to the intersection of the other two planes.
- Unique solution: \(x=0,\;y=5,\;z=3\)
- No solution: Inconsistent equations forming two parallel planes and one intersecting plane.
To form a system of equations to solve simultaneously, use the following steps:
- Carefully read the information provided.
- Organise the provided information into 'sets of information'.
- Identify the variables and assign them appropriate pronumerals.
- Form an equation for each set of information.
Example 5 – Forming a 2×2 system of equations
Two T-shirts and a pair of jeans cost $84. Five T-shirts and a pair of jeans cost $129.
Form two equations to represent this information.
| Step Instructions | Calculations |
|---|---|
| 1. Identify the sets of information provided in the scenario. |
|
| 2. Choose pronumerals to represent the variables and convert the sets of information into equations. |
Let t represent the cost of a T-shirt and j represent the cost of a pair of jeans.
|
Example 6 – Forming a 3×3 system of equations
A, B and C are three farmers who farm a combined total of 800 acres of land. Farmers A and B have a combined total of 543 acres between them, while Farmers B and C have a combined total of 498 acres between them.
Form a system of equations to represent this information.
| Step Instructions | Calculations |
|---|---|
| 1. Identify the sets of information provided in the scenario. |
|
| 2. Choose pronumerals to represent the variables and convert the sets of information into equations. |
Let A, B and C represent the three farmers.
|
Example 7 – Forming a 3×3 system of equations
The length, width and height of a room total 55 metres. The room is two metres wider than it is high and nine metres longer than it is wide.
Form a system of equations to represent this information.
| Step Instructions | Calculations |
|---|---|
| 1. Identify the sets of information provided in the scenario. |
|
| 2. Choose pronumerals to represent the variables and convert the sets of information into equations. |
Let L represent the length; W represent the width; and H represent the height of the room.
|
An understanding of the possible outcomes when attempting to solve simultaneous equations and what constitutes consistent and inconsistent equations allows us to answer questions in which a constant and/or a coefficient in one of the equations is undefined.
Example 8 – 2×2 Inconsistent equations
Example 9 – 2×2 Consistent (dependent) equations
Example 10 – 3×3 Inconsistent equations
Example 11 – 3×3 Consistent (dependent) equations
Example 12 – 3×3 Consistent (dependent) equations: A general case
Example 13 – 3×3 Consistent (dependent) equations: A general solution
- Coefficient
- Coincident
- Constant
- Dependent
- Elimination
- Equate
- Equation
- Geometric
- Inconsistent
- Intersect
- Multiple
- Parallel
- Plane
- Product
- Solution
- Solve
- Substitution
- Term
- Unique
- Undefined
- Unknown
- Variable
Character styles
Character styles are changes to the font of specific words and phrases.
Bold
Bold text is typically a darker, thicker version of the regular font. This is an example of bold text.
Use bold in the text body to emphasise individual terms or short phrases that are key knowledge points. In other words, use bold to draw the eye to the key terms and concepts in each section. It is best practice to use bold to highlight key terms the first time they appear in each topic.
To apply bold:
- highlight the word/phrase and either
- click the B in the wysiwyg toolbar or
- press ctrl + b.
- in Source, use the <strong></strong> tags around the characters you want to render as bold.
- For example, <strong>key term</strong>.
Spaces
This is a standard line of text with a straight paragraph style applied.
<p style="margin-left: 40px"> applies a style to the paragraph and indents the whole paragraph by 40 px. Lorem ipsum dolor sit amet, consectetur adipiscing elit. Donec enim libero, vehicula non maximus non, laoreet at justo. Fusce tempus non purus sodales iaculis.
<p style="text-indent: 40px"> applies a style to the paragraph and indents just the first line by 40 px. The following lines will be full width for the rest of the paragraph. Lorem ipsum dolor sit amet, consectetur adipiscing elit. Donec enim libero, vehicula non maximus non, laoreet at justo.
  or   creates a hair space [ ], which is very narrow.
  or   creates a thin space, which is slightly wider.
  forms a 3-per-em space [ ], which is between a thin and en-space.
  or   makes an en-space [ ], which is the width of the letter n.
  or   makes an em-space [ ], which is the width of the letter m.
CSS is a language that describes the style of an HTML document.
Subtraction symbol: - subtraction keyboard key
minus/negative symbol: − −
addition symbol: + addition keyboard key
plus/positive symbol: + +
| Triage Examination Form | ||||
|---|---|---|---|---|
| Demeanour:
BAR
QAR
Dull ✔
|
||||
| RR and effort: 40 bpm Moderate |
HR: 200 bpm |
Pulse: | MM / CRT: Pale |
Temperature: |
| Triage level:
Life-threatening ✔
Intermediate
Minor
Non-emergency
|
||||