- Check if a pair of numbers works as a solution for a set of equations
Determining if an Ordered Pair is a Solution to a System of Equations
A system of equations consists of two or more equations with the same variables. An ordered pair [latex](x, y)[/latex] is a solution to a system of equations if it satisfies all equations in the system simultaneously.
How to: Determine if an Ordered Pair is a Solution
- Substitute the [latex]x[/latex] and [latex]y[/latex] values of the ordered pair into both equations.
- Simplify each equation.
- Verify if both equations are true.
Consider the system of equations:
[latex]\begin{gathered}& 2x + y = 10 \\ & x - y = 2 \end{gathered}[/latex]
Let’s check if [latex](4, 2)[/latex] is a solution.
For equation 1: [latex]2x + y = 10[/latex] Substitute [latex]x = 4[/latex] and [latex]y = 2[/latex]:
[latex]2(4) + 2 = 10[/latex]
[latex]8 + 2 = 10[/latex]
[latex]10 = 10[/latex] (True)
For equation 2: [latex]x - y = 2[/latex] Substitute [latex]x = 4[/latex] and [latex]y = 2[/latex]:
[latex]4 - 2 = 2[/latex]
[latex]2 = 2[/latex] (True)
Since both equations are true when we substitute [latex](4, 2)[/latex], this ordered pair is a solution to the system.
Determine if the ordered pair [latex](3, -1)[/latex] is a solution to the following system of equations:
[latex]\begin{gathered}& 3x - 2y = 11 \\ & x + 4y = -1 \end{gathered}[/latex]
Determine if the ordered pair [latex](-2, 5)[/latex] is a solution to the system:
[latex]\begin{gathered}& 2x + 3y = 11 \\ & x - y = -7 \end{gathered}[/latex]
Determine if the ordered pair [latex](1, 2)[/latex] is a solution to the system:
[latex]\begin{gathered}& 4x - y = 2 \\ & x + 2y = 6 \end{gathered}[/latex]