How to Solve Simultaneous Equations

Question: What values of x and y solve the simultaneous equations below?

The equations can be solved by:

• Elimination.

• Substitution.

• Graphical methods.
Note: It is often useful to number the equations.

How to Solve Simultaneous Equations Using Elimination

Use the method of elimination to solve the simultaneous equations, Equations (1) and (2), at the top of the page.

Step 1
Check whether the coefficients of x or y are the same in both equations.

If Yes, skip to Step 3. If No go to Step 2.

Equation  coefficient of x   coefficient of y
The same?

Neither the coefficients of x nor y are the same. The answer to Step 1 is No, go to Step 2

Step 2
Make the coefficients of x or y the same by multiplying one or both of the equations. (Note: The new coefficient will be the least common multiple of the old coefficients).

Make the coefficients of x the same in Equation (1) and Equation (2).

The coefficients of x in the two equations are 2 and 1. The least common multiple of these is 2, so we need to multiply one or both equations to make the coefficients of x 2.

The coefficient of x in Equation (1) is already 2, so leave this equation as it is.

The coefficient of x in Equation (2) is 1, so multiply Equation (b) by 2.

Check whether the coefficients of x or y are the same now:

Equation  coefficient of x   coefficient of y
The same?

Step 3
Eliminate the unknown with the same coefficient by either adding or subtracting the two equations.

x has the same coefficient in both equations. Subtract Equation (2) from Equation (1) to eliminate the x:

Note: The minus applies to all of Equation (2) when subtracting. 2x - 2x, y - 6y, 4 - 14.

Step 4
Solve the equation obtained in Step 3 to find the unknown (y):

Step 5
Substitute the value from Step 4 (y = 2) into either of the original equations and solve:

Substitute y = 2 into Equation (1):

Step 6
Solve this equation to find the remaining unknown (x):

The solution to the simultaneous equation is:

The slider below shows an example of solving simultaneous equations using the elimination method: Read more about the elimination method

How to Solve Simultaneous Equations Using Substitution

Use the method of elimination to solve the simultaneous equations, Equations (1) and (2), at the top of the page.

Step 1
Rearrange one of the simultaneous equations to make either x or y the subject of the equation.

Rearrange Equation (1) to make x the subject:

Step 2
Substitute the equation obtained in Step 1 into the other simultaneous equation (the one which was not rearranged).

Step 3
Solve this equation to find the unknown (y):

Step 4
Substitute the value from Step 3 (y = 2) into either of the original equations and solve:

Substitute y = 2 into Equation (1):

Solving:

The solution to the simultaneous equation is:

The slider below shows an example of solving simultaneous equations using the substitution method: Read more about the substitution method

How to Solve Simultaneous Equations Using Graphical Methods

Use graphical methods to solve the simultaneous equations, Equations (1) and (2), at the top of the page.

Step 1
Plot Equation (1) as a line on a pair of axes:

Step 2
Plot Equation (2) as a line on the same pair of axes:

Step 3
Find the point where the two lines cross.

Step 4
The x-coordinate of the point found in Step 3 gives the value of x that solves the simultaneous equations. (Note: Find this by drawing a line straight down to the x-axis).

x = 1 is the value of x that solves the simultaneous equations.

Step 5
The y-coordinate of the point found in Step 3 gives the value of y that solves the simultaneous equations. (Note: Find this by drawing a line straight across to the y-axis).

y = 2 is the value of y that solves the simultaneous equations.

The solution to the simultaneous equation is:

The slider below shows an example of solving simultaneous equations using the graphical method: Read more about the graphical method
show

Note
WHAT ARE coefficients?

The coefficients of x and y are the numbers in front of the x and y in the equation.

Note: When x or y appear on their own, without a number in front of it, its coefficient is 1.

Note
WHAT IS THE LEAST COMMON MULTIPLE?

A multiple of a number is the result of multiplying a number by an integer.

To find the least common multiple of two numbers (such as the coefficients of x: 1, 2):

• List the multiples of the two numbers:

• Find the least multiple that is the same (common) in both lists:

• 2 is the least common denominator.

CAREFUL WHEN ADDING AND SUBTRACTING EQUATIONS

coefficients in the equations can be positive and negative, so care is needed when adding or subtracting equations.

Consider the simultaneous equations below:

• To eliminate the x, subtract Equation (2) from Equation (1).

• Subtract term by term:

This leaves:

HOW DO I KNOW WHETHER TO ADD OR SUBTRACT?

What if you wanted to eliminate the y from the simultaneous equations?

The equations must be added as:

In general:

• If the signs of the coefficients are different, add the equations.

• If the signs of the coefficients are the same, subtract the equations.
Note
HOW DO I PLOT THE EQUATIONS The equations in the simultanous equations are linear equations. When they are plotted they form straight lines.

The standard form for the equation of a straight line is:

where m is the slope of the line, and c is where the line crosses the y-axis.

One way of plotting the equations is to rearrange them into this form. For example, consider the simultaneous equations below:

Rearranging the equations into the standard form for straight lines gives:

These lines can then plotted, just find out what values of y you get for different values of x:

These points can then be plotted on a pair of x-y axes.

Alternatively,

• Find where the line crosses the y-axis by inserting x = 0 into the equation:

Plot this point on a pair of x-y-axes:
• Find where the line crosses the x-axis by inserting y = 0 into the equation:

Plot this point on the same pair of axes and join the two points with a straight line:

• Note: To find where it crosses the y-axis, put x = 0.

To find where it crosses the x-axis, put y = 0.