# How to Solve Simultaneous Equations (Mathematics Lesson)

# 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

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

**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

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

**Step 3**to find the unknown (y):

Step 5

**Step 4**(y = 2) into either of the original equations and solve:

Substitute y = 2 into Equation (1):

Step 6

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 Equation (1) to make x the subject:

Step 2

**Step 1**into the other simultaneous equation (the one which was not rearranged).

Step 3

Step 4

**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

Step 2

Step 3

Step 4

**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

**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

##### Interactive Test

**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:

**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**.