Systems of Equations

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?

Consider $x+y=6$`x`+`y`=6. This could have the solution $x=2$`x`=2 & $y=4$`y`=4, or another solution of $x=40$`x`=40 & $y=-34$`y`=−34. In fact, there are an infinite number of solutions to this equation.

If we wanted to pick out a single solution to this equation, we'd have to describe some extra condition that narrows down the infinite number of solutions to just one. Often, we can do this by considering a second equation involving the same variables, together with the first equation.

For example, if we wanted to a solution to $x+y=6$`x`+`y`=6 where the value of $x$`x` is $5$5, then we would be looking for a pair of values for $x$`x` and $y$`y` that are solutions to the equation $x+y=6$`x`+`y`=6 and also to the equation $x=5$`x`=5. Solving the first equation when $x$`x` is equal to $5$5, we can find that the value of $y$`y` must be $1$1.

On the number plane, this looks like the the point of intersection between the lines $x+y=6$`x`+`y`=6 and $x=5$`x`=5.

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 |
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$x$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 |
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$x$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.

Consider the equations $y=2x$`y`=2`x` and $y=28-2x$`y`=28−2`x`, which have the following tables of values.

Find the values for $x$`x` and $y$`y` which satisfy the equations $y=2x$`y`=2`x` and $y=28-2x$`y`=28−2`x` simultaneously.

$y=2x$y=2x |
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$x$x |
$3$3 | $4$4 | $5$5 | $6$6 | $7$7 |
---|---|---|---|---|---|

$y$y |
$6$6 | $8$8 | $10$10 | $12$12 | $14$14 |

$y=28-2x$y=28−2x |
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$x$x |
$3$3 | $4$4 | $5$5 | $6$6 | $7$7 |
---|---|---|---|---|---|

$y$y |
$22$22 | $20$20 | $18$18 | $16$16 | $14$14 |

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

`x`=,`y`=

Use the table of values to find the pair of $x$`x` and $y$`y` values that satisfy both $x+y=5$`x`+`y`=5 and $y=2x-1$`y`=2`x`−1.

Complete the tables.

$x+y=5$ `x`+`y`=5$x$ `x`$1$1 $2$2 $3$3 $y$ `y`$\editable{}$ $\editable{}$ $\editable{}$ $y=2x-1$ `y`=2`x`−1$x$ `x`$1$1 $2$2 $3$3 $y$ `y`$\editable{}$ $\editable{}$ $\editable{}$ What values of $x$

`x`and $y$`y`satisfy both $x+y=5$`x`+`y`=5 and $y=2x-1$`y`=2`x`−1?$x=\editable{},y=\editable{}$

`x`=,`y`=

Dave obtained quotes from two plumbers.

Plumber $A$`A` charges $\$92$$92 for a callout fee plus $\$17$$17 per hour.

Plumber $B$`B` charges $\$20$$20 for a callout fee plus $\$35$$35 per hour.

Complete the table of values.

Hours of work Amount charged by plumber $A$ `A`Amount charged by plumber $B$ `B`$4$4 $\editable{}$ $\editable{}$ $5$5 $\editable{}$ $\editable{}$ $6$6 $\editable{}$ $\editable{}$ $7$7 $\editable{}$ $\editable{}$ How many hours would the job have to take for plumber $A$

`A`and plumber $B$`B`to charge the same amount?

Solve systems of two linear equations involving two variables with integer coefficients, using the algebraic method of substitution or elimination