Introduction
Solving a single linear equation is when we solve for one variable, such as x. For example, rearranging x+2=5 will create a unique answer of 3 for x. So what happens when there is more than one variable in an equation?
Ideas
Graphical method
Consider the equation x + y = 6. This could have the solution x=2 , y = 4, or it could have the solution of x = 40, y =-34. In fact, there are infinitely many solutions to this equation. Just choose any value for x, and then use the equation to find the corresponding value of y required to make the equation hold true.
If we have two equations with the same two variables in them (such as x and y), we call the set of equations system of equations. They are also commonly referred to as simultaneous equations.
To solve a system of linear equations is to find a common pair of x and y values that satisfies both of these equations simultaneously. If there is a single pair of values of x and y that successfully does this, then the system of equations has a unique solution.
If the lines are parallel and distinct, then there will be no points of intersection between them. That is, there are no corresponding x- and y-values that satisfy both equations simultaneously.
The final case to consider is when two different equations have the same graphical representation. For example, if the graphs of x+y=5 and 2x + 2y = 10 were placed on the same set of axes, we would end up with two lines lying perfectly on top of one another. Every point on that line will satisfy both equations and is a point of intersection, meaning there are infinitely many solutions to this type of system of equations.
Examples
Example 1
The following graph displays a system of two equations. Write down the solution to the system in the form (x, y).
If the straight lines representing the two equations are not parallel, then there will be exactly one point of intersection between them.
If the lines are parallel and distinct, then there will be no points of intersection between them.
If two different equations have lines that lie on top of each other, then there will be infinitely many solutions.
Substitution method
One way to solve simultaneous equations is called the substitution method. In this chapter, the simultaneous equations that will be mostly studied involve 2 variables and 2 equations. In these cases, the substitution method works by solving one variable first through 'substituting' one equation into the other.
Name the equations by writing (1) and (2) next to them, and whenever we create new equations out of one or both of them, we can name them (3), (4), and so on. This way, when substituting equation (3) into equation (1), we can write it in shorthand as (3) → (1). This helps to keep everything order.
When dealing with equations with big numbers, see if they can be simplified before beginning to solve them. For example, 2x - 4 = 6y can be simplified to x - 2 = 3y without changing the values of x and y.
Remember to solve for the values of both variables. Check that you have both parts of the solution at the end of every simultaneous equation problem, unless you are specifically solving for just one of the variables.
Examples
Example 2
We want to solve the following system of equations using the substitution method.
Equation 1: \, \, y = -2x - 1
Equation 2: \, \, x - 6y = -59
First solve for x.
Now solve for y.
The substitution method involves substituting an expression for one variable from one of the equations into the other equation. Then we can solve this new equation for the other variable.
This method is most useful when one of the equations has a variable as the subject.
Elimination method
This is another method of solving simultaneous equations. It works by adding or subtracting equations from one another to eliminate one variable. For simultaneous equations with 2 variables and 2 unknowns that have a unique solution, the elimination method will create 1 linear equation with 1 unknown variable, which can then be solved.
Sometimes two simultaneous equations will not have the same value coefficients for any of the 2 variables that need to be eliminated. In this case we can multiply or divide the whole equation by a constant until both equations have 1 variable with matching coefficients.
Examples
Example 3
Use the elimination method by adding both equations to solve for x first and then y.
Equation 1: \, \, 2x + 5y = 44
Equation 2: \, \, 6x - 5y = -28
Solve for x.
Now find the value of y.
Example 4
Use the elimination method to solve for x and y.
Equation 1: \, \, -6x - 2y = 46
Equation 2: \, \, -30x - 6y = 246
First solve for x.
Now solve for y.
The elimination method involves adding or subtracting equations from one another to eliminate one variable, so we can solve for the other variable.
We should add the equations when the coefficients of the variable we want to eliminate are equal but opposite in sign, and subtract the equations when they are equal and the same sign.
To get the same coefficients of a variable, we can multiply or divide the whole equation by a constant.
Solve simultaneous equations using a CAS calculator
A CAS calculator can be utilised in two different ways to solve a linear system of equations. A graphical approach would be to use a CAS calculator to graph both linear equations on the same set of axis, then use the CAS calculator to find the point of intersection.
Alternatively, the solve function of the CAS calculator can be used to find the point of intersection immediately with no graphing involved.
Examples
Example 5
Consider the system of equations below.
Equation 1: \, \, 3x-1.6y=5.9
Equation 2: \, \, 0.45x+0.7y=-0.02
Solve the system of linear equations using the solving functionality of your CAS calculator, leaving your answer as a pair of coordinates, correct to 2 decimal places.
A CAS calculator can be utilised in two different ways to solve a linear system of equations. A graphical approach would be to use a CAS calculator to graph both linear equations on the same set of axis, then use the CAS calculator to find the point of intersection.