Is (x, y) a solution to the pair of equations?

key notes:

A point is a solution to simultaneous equations if plugging the point into each equation results in a true statement.

Systems of equations :

A system of equations consists of two or more equations with the same variables. For example, the linear equations below create a system of linear equations:

  • y=2x–8
  • 3x–4y=7

A solution to a system of equations is a set of values that makes each equation in the system true. The solution to the system of equations above can be written as the ordered pair (5,2) because x=5 and y=2 satisfy both equations in the system. To show this, you can substitute 5 for x and 2 for y into both equations.

  • y=2x–8
  • 2=2(5)–8
  • 2=10–8
  • 2=2
  • 3x–4y=7
  • 3(5)–4(2)=7
  • 15–8=7
  • 7=7

Learn with an example

Is (2, 1) a solution to these simultaneous equations?

x + 14y = 16
2x + 5y = 9

In the ordered pair (2, 1), 2 is the x-value and 1 is the y-value.

In the first equation, replace x with 2 and y with 1.

x + 14y = 16

2 + 14(1) = 16

2+ 14 = 16

16 = 16

Yes, 16 = 16.

Now in the second equation, replace x with 2 and y with 1.

2x + 5y = 9
2(2) + 5(1) = 9

4 + 5 = 9

9 = 9

Yes, 9 = 9.

Plugging (2, 1) into each equation resulted in a true statement. So, (2, 1) is a solution to the simultaneous equations.

Is (1, 1) a solution to these simultaneous equations?

x + 4y = 14
2x + 18y = 20

  • yes
  • no

In the ordered pair (1, 1), the x-value and the y-value are both 1.

In the first equation, replace x and y with 1.

x + 4y = 14

1 + 4(1)?14

1 + 4?14

5?14

No, 5 ≠ 14.

Plugging (1, 1) into the first equation did not result in a true statement. So, (1, 1) is not a solution to the simultaneous equations.

Is (3, 9) a solution to these simultaneous equations?

y = 9x + 2
y = x + 6

  • yes
  • no

In the ordered pair (3, 9), 3 is the x-value and 9 is the y-value.

In the first equation, replace x with 3 and y with 9.

y = 9x + 2

9? 9(3) + 2

9?27 + 2

9?29

No, 9 ≠ 29.

Plugging (3, 9) into the first equation did not result in a true statement. So, (3, 9) is not a solution to the simultaneous equations.

Let’s practise: