# Solutions to Equations and Inequalities

A **solution** to an equation assigns a value to each variable in the equation, and these values satisfy the equation. To **satisfy** an equation means to make the equation a true statement.

\[\begin{array}{rclcc} 2x-3y & = & 1; & x = 2; & y=1 \\ & \Downarrow & & & \\ 2(2)-3(1) & = & 1 & & \\ 1 & = & 1 & & \end{array}\]

The values \(x = 2\) and \(y=1\) satisfy the equation \(2x-3y=1\), so they are a solution to the equation.

A **solution** has a similar meaning for inequalities, systems of equations and other mathematical statements — the solution assigns values to each variable, and these values satisfy the mathematical statements. In some cases, there may be more than one solution, or there may be no solutions.

## Verifying Solutions

Verifying solutions

## Solutions to Systems of Equations

In a system of linear equations, each equation typically represents some kind of constraint on the problem. The **solution** to a system of equations will assign a value to each variable in the system, and these values will satisfy every equation in the system.

\[\begin{align} 3x+y &= 5 \\ x-y &= -1 \end{align}\]

Is \(x=1,y=2\) a solution to the system above?

Assign the given values to the equations to see if they satisfy the equations:\[\begin{align} 3x+y &= 5 \\ 3(1)+(2) &\stackrel{?}{=} 5 \\ 5 &= 5 \\ \\ x-y &= -1 \\ (1)-(2) &\stackrel{?}{=} -1 \\ -1 &= -1 \end{align}\]

Both equations are satisfied, so \(x = 1,y=2\) is a solution.

Keep in mind that a solution must satisfy every equation in the system in order to be a solution.

\[\begin{align} 3x+y &= 5 \\ x-y &= -1 \end{align}\]

Is \(x=0,y=5\) a solution to the system above?

Assign the given values to the equations to see if they satisfy the equations:\[\begin{align} 3x+y &= 5 \\ 3(0)+(5) &\stackrel{?}{=} 5 \\ 5 &= 5 \\ \\ x-y &= -1 \\ (0)-(5) &\stackrel{?}{=} -1 \\ -5 &\not= -1 \end{align}\]

Although the first equation is satisfied, the second equation is not. This means that \(x =0,y=5\) is not a solution.

It is possible that there are infinitely many solutions for a system.

[Example of a given system that has infinitely many solutions]]

It is also possible that a system of equations has no solutions.

[Example of a given system that has no solutions]]

[2 example problems about identifying solutions]

**Cite as:**Solutions to Equations and Inequalities.

*Brilliant.org*. Retrieved from https://brilliant.org/wiki/solutions-to-equations/