4.01 Solutions to linear systems using a table of values
So far we've only had to solve equations with one variable. We know that rearranging $x+2=5$x+2=5 will give us a unique answer of $3$3 for $x$x, but what happens when we have more than one variable in our equation?
Exploration
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 infinitely many solutions to this equation. Just pick any value for $x$x, and then use the equation to find the corresponding value of $y$y required to make the equation hold true.
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 a values of $x$x and $y$y that successfully do this, then we will have found a unique solution to our system.
Graphical representation
On a coordinate plane, the solution is represented by the point of intersection of the two equations' graphs (where the two graphs cross over). So the $x$x and $y$y values of our 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 each set of two simultaneous linear equations?
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.
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.
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$2x+2y=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.
So where might this system of equations come from, and why is it important to know how many solutions it has?
Worked example
example 1
Leah got a quote from two photographers for an event. Photographer $A$A charges a $\$48$$48 booking fee plus $\$17$$17 per hour of the event, whilst photographer $B$B has a $\$28$$28 booking fee and charges $\$21$$21 per hour. Form a system of equations that represents this information, and determine how many solutions the system will have.
Think: Which quantities from this situation will the $x$x and $y$y represent? Are any of them related to one another?
Do: The duration of the event is variable and will affect the overall cost of the photographer, so lets represent the duration of the event with $x$x and the total cost of the photographer for the event with $y$y. Then we could form one equation relating the duration and the cost of the event for each photographer as follows:
| $y$y | $=$= | $48+17x$48+17x | $(A)$(A) |
| $y$y | $=$= | $28+21x$28+21x | $(B)$(B) |
We've now formed an appropriate system of equations, so how can we tell how many solutions it will have?
Looking at the coefficient of $x$x in each equation tells us the lines representing them have different gradients, so looking at the green summary box from earlier in the lesson, we know the system will have exactly one solution.
Reflect: Since the lines representing these cost equations have a point of intersection, this means there will be a duration of the event for which the photographers cost the same. More interestingly, this implies that the cheapest option for photographer will depend on the length of the event.
Practice questions
Question 1
How can we know whether a given ordered pair is a solution of a system of equations?
If substituting the pair into each equation results in zero.
AIf substituting the pair into each equation makes the left-hand side equal to the right-hand side.
BIf substituting the pair into each equation eliminates the variables.
C
Question 2
We are going to determine whether the point $\left(2,-2\right)$(2,−2) is a solution of the system of equations:
$2x+3y=-2$2x+3y=−2
$4x+3y=5$4x+3y=5
Using the first equation, $2x+3y=-2$2x+3y=−2, find the value of $y$y when $x=2$x=2.
Now using the second equation, $4x+3y=5$4x+3y=5, find the value of $y$y when $x=2$x=2.
Hence, is $\left(2,-2\right)$(2,−2) a solution of the system?
Yes
ANo
B
Question 3
Gwen has a total of $\$11500$$11500, which she has put into two accounts. She has four times as much money in her savings account as in her checking account.
Let $x$x represent the amount in her checking account and let $y$y represent the amount in her savings account.
Write a system of two equations that describes all the information provided, giving both equations on the same line separated by a comma. You do not need to solve the system.
Solving systems of equations
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.
Worked example
example 2
Consider the system of equations $y=2x+1$y=2x+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.
Practice questions
Question 4
Consider the equations $y=2x$y=2x and $y=28-2x$y=28−2x, which have the following tables of values.
Find the values for $x$x and $y$y which satisfy the equations $y=2x$y=2x and $y=28-2x$y=28−2x simultaneously.
| $y=2x$y=2x | |||||
| $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 | |||||
| $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=
Question 5
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=2x−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=2x−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=2x−1?
$x=\editable{},y=\editable{}$x=,y=
Question 6
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?