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

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?

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Show Answer

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?

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Show Answer

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]