The Hiking Club is buying nuts to make trail mix for a fundraiser. Three pounds of almonds and two pounds of cashews cost a total of $36. Three pounds of cashews and two pounds of almonds cost a total of $39. Is (a, c) = ($6, $9) a solution to this system?
Solution to a System of Linear Equations
A system of linear equations consists of the equations of two lines.The solution to a system of linear equations is the point which lies on both lines. In other words, the solution is the point where the two lines intersect. To verify whether a point is a solution to a system or not, we will either determine whether it is the point of intersection of two lines on a graph or we will determine whether or not the point lies on both lines algebraically.
Let's determine whether the given points are a solution to the systems of equations below.
 Is the point (5, 2) the solution of the system of linear equations shown in the graph below?
Yes, the lines intersect at the point (5, 2) so it is the solution to the system.
 Is the point (3, 4) the solution to the system given below?
\begin{align*}2x3y &= 18\\ x+2y &= 6\end{align*}
No, (3, 4) is not the solution. If we replace the \begin{align*}x\end{align*} and \begin{align*}y\end{align*} in each equation with 3 and 4 respectively, only the first equation is true. The point is not on the second line; therefore it is not the solution to the system.
Now, let's find the solution to the system below.
\begin{align*} x &= 5 \\ 3x2y &= 25\end{align*}
Because the first line in the system is vertical, we already know the xvalue of the solution, \begin{align*}x=5\end{align*}. Plugging this into the second equation, we can solve for y.
\begin{align*}3(5)2y &= 25 \\ 152y &= 25 \\ 2y &= 10 \\ y &= 5 \end{align*}
The solution is (5, 5). Check your solution to make sure it's correct.
\begin{align*}3(5)2(5) &= 25 \\ 15 + 10 &= 25\end{align*}
You can also solve systems where one line is horizontal in this manner.
Examples
Example 1
Earlier, you were asked if (a, c) = ($6, $9) is a solution to the system of equations.
The system of linear equations represented by this situation is:
\begin{align*}3a + 2c &= 36\\ 3c + 2a &= 39\end{align*}
If we plug in $6 for a and $9 for c, both equations are true. Therefore ($6, $9) is a solution to the system.
Example 2
Is the point (2, 1) the solution to the system shown below?
No, (2, 1) is not the solution. The solution is where the two lines intersect which is the point (3, 1).
Example 3
Verify algebraically that (6, 1) is the solution to the system shown below.
\begin{align*}3x4y &= 22\\ 2x+5y &= 7\end{align*}
By replacing \begin{align*}x\end{align*} and \begin{align*}y\end{align*} in both equations with 6 and 1 respectively (shown below), we can verify that the point (6, 1) satisfies both equations and thus lies on both lines.
\begin{align*}3(6)4(1) &= 18+4=22\\ 2(6)+5(1) &= 125=7\end{align*}
Example 4
Explain why the point (3, 7) is the solution to the system:
\begin{align*}y &= 7\\ x &= 3\end{align*}
The horizontal line is the line containing all points where the \begin{align*}y\end{align*}coordinate 7. The vertical line is the line containing all points where the \begin{align*}x\end{align*}coordinate 3. Thus, the point (3, 7) lies on both lines and is the solution to the system.
Review
Match the solutions with their systems.
 (1, 2)
 (2, 1)
 (1, 2)
 (1, 2)
Determine whether each ordered pair represents the solution to the given system.
 .

 \begin{align*}4x+3y &= 12\\ 5x+2y &= 1; \ (3, 8)\end{align*}
 .

 \begin{align*}3xy &= 17\\ 2x+3y &= 5; \ (5, 2)\end{align*}
 .

 \begin{align*}7x9y &= 7\\ x+y &= 1; \ (1, 0)\end{align*}
 .

 \begin{align*}x+y &= 4\\ xy &= 4; \ (5, 9)\end{align*}
 .

 \begin{align*}x &= 11\\ y &= 10; \ (11, 10)\end{align*}
 .

 \begin{align*}x+3y &= 0\\ y &= 5; \ (15, 5)\end{align*}
Find the solution to each system below.
 .

 \begin{align*}x &= 2\\ y &= 4 \end{align*}
 .

 \begin{align*}y &= 1\\ 4x  y &= 13 \end{align*}
 .

 \begin{align*}x &= 7\\ y &= 6 \end{align*}
 .

 \begin{align*}x &= 2\\ 8x+3y &= 11\end{align*}
 Describe the solution to a system of linear equations.
 Can you think of why a linear system of two equations would not have a unique solution?
Answers for Review Problems
To see the Review answers, open this PDF file and look for section 3.1.