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# Checking a Solution for a Linear System

## Substitute given values or graph to compare coordinates to intersections

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Checking a Solution for a Linear System

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.

1. 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.

1. Is the point (-3, 4) the solution to the system given below?

2x3yx+2y=18=6\begin{align*}2x-3y &= -18\\ x+2y &= 6\end{align*}

No, (-3, 4) is not the solution. If we replace the x\begin{align*}x\end{align*} and y\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.

x3x2y=5=25\begin{align*} x &= 5 \\ 3x-2y &= 25\end{align*}

Because the first line in the system is vertical, we already know the x-value of the solution, x=5\begin{align*}x=5\end{align*}. Plugging this into the second equation, we can solve for y.

3(5)2y152y2yy=25=25=10=5\begin{align*}3(5)-2y &= 25 \\ 15-2y &= 25 \\ -2y &= 10 \\ y &= -5 \end{align*}

The solution is (5, -5). Check your solution to make sure it's correct.

3(5)2(5)15+10=25=25\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:

3a+2c3c+2a=36=39\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.

3x4y2x+5y=22=7\begin{align*}3x-4y &= 22\\ 2x+5y &= 7\end{align*}

By replacing x\begin{align*}x\end{align*} and y\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.

3(6)4(1)2(6)+5(1)=18+4=22=125=7\begin{align*}3(6)-4(-1) &= 18+4=22\\ 2(6)+5(-1) &= 12-5=7\end{align*}

#### Example 4

Explain why the point (3, 7) is the solution to the system:

yx=7=3\begin{align*}y &= 7\\ x &= 3\end{align*}

The horizontal line is the line containing all points where the y\begin{align*}y-\end{align*}coordinate 7. The vertical line is the line containing all points where the x\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. (1, 2)
1. (2, 1)
1. (-1, 2)
1. (-1, -2)

Determine whether each ordered pair represents the solution to the given system.

1. .
4x+3y5x+2y=12=1; (3,8)\begin{align*}4x+3y &= 12\\ 5x+2y &= 1; \ (-3, 8)\end{align*}
1. .
3xy2x+3y=17=5; (5,2)\begin{align*}3x-y &= 17\\ 2x+3y &= 5; \ (5, -2)\end{align*}
1. .
7x9yx+y=7=1; (1,0)\begin{align*}7x-9y &= 7\\ x+y &= 1; \ (1, 0)\end{align*}
1. .
x+yxy=4=4; (5,9)\begin{align*}x+y &= -4\\ x-y &= 4; \ (5, -9)\end{align*}
1. .
xy=11=10; (11,10)\begin{align*}x &= 11\\ y &= 10; \ (11, 10)\end{align*}
1. .
x+3yy=0=5; (15,5)\begin{align*}x+3y &= 0\\ y &= -5; \ (15, -5)\end{align*}

Find the solution to each system below.

1. .
xy=2=4\begin{align*}x &= -2\\ y &= 4 \end{align*}
1. .
y4xy=1=13\begin{align*}y &= -1\\ 4x - y &= 13 \end{align*}
1. .
xy=7=6\begin{align*}x &= 7\\ y &= 6 \end{align*}
1. .
x8x+3y=2=11\begin{align*}x &= 2\\ 8x+3y &= -11\end{align*}
1. Describe the solution to a system of linear equations.
2. Can you think of why a linear system of two equations would not have a unique solution?

To see the Review answers, open this PDF file and look for section 3.1.

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