5. Systems of Equations & Inequalities

Lesson

We often solve equations that involve only one variable. For example, we can rearrange the equation $x+2=5$`x`+2=5 to give us a unique answer of $3$3 for $x$`x`. But, what happens when we have more than one variable in our equation?

If we have **two **equations with the same two variables in them ($x$`x` and $y$`y`), then we call them a system of equations. They are also commonly referred to as simultaneous equations.

We might be interested in finding a common pair of $x$`x` and $y$`y` values that satisfies both of these equations simultaneously. If we can find any values of $x$`x` and $y$`y` that successfully do this, then we will have found a unique solution to our system. Like in the above example, this unique solution can be represented by the intersection of two graphs on the number plane.

One way to find solutions common to each equation is to complete a table of values for each equation and to see if there are any common pairs of values.

Consider the system of equations $y=2x+1$`y`=2`x`+1 and $x+y=4$`x`+`y`=4. Let's find values for $x$`x` and $y$`y` which satisfy both equations simultaneously.

First, we will need to write up the table of values for each of the equations in the system:

$y=2x+1$y=2x+1 |
|||||

$x$x |
$0$0 | $1$1 | $2$2 | $3$3 | $4$4 |
---|---|---|---|---|---|

$y$y |
$1$1 | $3$3 | $5$5 | $7$7 | $9$9 |

$x+y=4$x+y=4 |
|||||

$x$x |
$0$0 | $1$1 | $2$2 | $3$3 | $4$4 |
---|---|---|---|---|---|

$y$y |
$4$4 | $3$3 | $2$2 | $1$1 | $0$0 |

Are there any pairs of $x$`x` and $y$`y` values that are common between these two tables? Yes! Looking at the column where $x=1$`x`=1, we can see that both equations are satisfied when it is also the case that $y=3$`y`=3.

So, the solution to this system of equations is $x=1$`x`=1 & $y=3$`y`=3.

How can we know whether a given ordered pair is a solution of a system of equations? Select all that apply.

If substituting the pair into each equation results in a true statement.

AIf substituting the pair into each equation eliminates the variables.

BIf substituting the pair into each equation results in zero.

CIf substituting the pair into each equation makes the left-hand side equal to the right-hand side.

DIf substituting the pair into each equation results in a true statement.

AIf substituting the pair into each equation eliminates the variables.

BIf substituting the pair into each equation results in zero.

CIf substituting the pair into each equation makes the left-hand side equal to the right-hand side.

D

We are going to determine whether the point $\left(5,2\right)$(5,2) is a solution of the system of equations:

$x+y=7$`x`+`y`=7

$x-y=3$`x`−`y`=3

Using the first equation, $x+y=7$

`x`+`y`=7, find the value of $y$`y`when $x=5$`x`=5.Now using the second equation, $x-y=3$

`x`−`y`=3, find the value of $y$`y`when $x=5$`x`=5.Is $\left(5,2\right)$(5,2) a solution of the system?

Yes

ANo

BYes

ANo

B

On a coordinate plane, the solution is represented by the point of intersection (where the two graphs cross over) of the two equations' graphs. So the $x$`x` and $y$`y` values of the solution will take the form of coordinates of the intersection point $\left(x,y\right)$(`x`,`y`).

With this in mind, how many solutions should be possible for a system of linear equations?

Number of Solutions | 1 | 0 | Infinitely many |
---|---|---|---|

Equations | $y=\frac{x}{2}-2$y=x2−2$y=-x+7$ y=−x+7 |
$y=\frac{x}{2}-2$ |
$y=\frac{x}{2}-2$y=x2−2$y=\frac{x}{2}-2$ y=x2−2 |

Graph |

Number of solutions to a system

**One solution: **If the lines representing the two equations are not parallel, then there should be **exactly one** point of intersection between them (as pictured above). Have a think about why this is true.

**No solution:** If the lines are parallel and distinct, then there **will not** be any points of intersection between them. This means no corresponding $x$`x` and $y$`y` values satisfy both equations simultaneously.

**Infinitely many solutions: **The final case to consider is when two different equations have the same graphical representation. For example, if the graphs of $x+y=5$`x`+`y`=5 and $2x+2y=10$2`x`+2`y`=10 were placed on the same set of axes, we would end up with two lines lying perfectly on top of one another. So every point on the line is a point of intersection, meaning there is **infinitely many** solutions to this system of equations.

Not all systems of linear equations have a solution. Let's think about when systems of linear equations will not have a solution graphically. It happens when they're parallel!

Let's remind ourselves that to find the slope of a linear equation all we have to do is put it in the slope-intercept form $y=mx+b$`y`=`m``x`+`b` and $m$`m` will be our slope. This means that for example, the system of linear equations $y=3x-1$`y`=3`x`−1 and $y=3x+6$`y`=3`x`+6 will never have a solution since they both have a slope of $3$3.

The following graph displays a system of two equations.

Loading Graph...

How many solutions does this system of equations have?

Infinitely many solutions.

ANo solutions.

BOne solution.

CInfinitely many solutions.

ANo solutions.

BOne solution.

CWrite down the solution to the system of equations as an ordered pair $\left(x,y\right)$(

`x`,`y`).

Consider the following linear equations:

$y=5x-7$`y`=5`x`−7 and $y=-x+5$`y`=−`x`+5

Plot the lines of the 2 equations on the same graph.

Loading Graph...State the values of $x$

`x`and $y$`y`which satisfy both equations.$x$

`x`= $\editable{}$$y$

`y`= $\editable{}$

The following graph displays a system of two equations.

Loading Graph...

How many solutions does this system of equations have?

No solutions

AOne solution

BInfinitely many solutions

CNo solutions

AOne solution

BInfinitely many solutions

C

Represent, solve and interpret equations/inequalities and systems of equations/inequalities algebraically and graphically.

Write and/or solve a system of linear equations (including problem situations) using graphing, substitution, and/or elimination. Note: Limit systems to two linear equations.