A linear equation is one in which the variables are to the power one. Equations like the following are typical examples.
$2x+\frac{1}{2}y=14$2x+12y=14
$y=-3x+7$y=−3x+7
$x_1+x_2-x_3=0$x1+x2−x3=0
A solution to any one of these equations is a set of values of the variables that satisfies the equation. For example, the equation $2x+\frac{1}{2}y=14$2x+12y=14 has solutions $(x,y)=(7,0),(5,8),(3,16),(21,-56)$(x,y)=(7,0),(5,8),(3,16),(21,−56) and countless others.
The second equation, $y=-3x+7$y=−3x+7 also has infinitely many solutions. Among them is the solution $(x,y)=(21,-56)$(x,y)=(21,−56). This pair of numbers also satisfies the first equation. We say it is a simultaneous solution to the two equations.
The third equation has solutions with three numbers. For example, $(x_1,x_2,x_3)=(2,1,3),(5,-1,4),(21,-56,-35)$(x1,x2,x3)=(2,1,3),(5,−1,4),(21,−56,−35), and so on. Again, there are infinitely many solutions.
If the equation has two variables, $x$x and $y$y, say, then the relation between them can be displayed as a $2$2-dimensional graph. The graphs of linear equations in two variables are straight lines. A graph contains all the points that satisfy the equation.
If there are three variables, the graph is in $3$3-dimensions and is in the form of a plane.
Solutions to systems of equations correspond to intersection points on their graphs. Such points are on both graphs and satisfy both equations.
If two graphs intersect at one point, then there is a unique simultaneous solution to the pair of equations. This is the case for the pair
Here, the two graphs are shown. The intersection is at the point $(21,-56)$(21,−56).
It can happen that the graphs of two equations coincide. In other words, all the points that satisfy one equation also satisfy the other. Thus, there are infinitely many solutions. This would be the case for the equations $y=-3x+7$y=−3x+7 and $2y+6x-14=0$2y+6x−14=0.
These equations look different as written but they express the same relation and each can be converted into the other by algebraic manipulation.
The possibility of having infinitely many solutions takes on more importance in the case of systems of equations with three variables. These describe planes and two different planes can intersect in a line so that infinitely many solutions exist, all of them lying on that line.
It can also happen that a pair of lines does not intersect. Hence, a pair of equations describing the lines has no simultaneous solution. Such equations are said to be inconsistent. For example, the graphs of $y=-3x+7$y=−3x+7 and $y=-3x+28$y=−3x+28 do not intersect. Their slopes are the same and one is a vertical translation of the other. The graphs are shown below.
If two equations arranged into the same form have the same coefficients for the variables but different constant terms, then they cannot be true simultaneously. No solution exists and the graphs are parallel. For example, the following are inconsistent:
$2x+5y=19$2x+5y=19 and $2x+5y=16$2x+5y=16
$y=-x+5$y=−x+5 and $y=-x+6$y=−x+6
Example 1
One quantity is $1.3$1.3 more than another and together they make $3.7$3.7. What are the two quantities?
We must first express the facts algebraically. Call the quantities $x$x and $y$y. Then, we have $y=x+1.3$y=x+1.3 and $x+y=3.7$x+y=3.7.
One way to find the simultaneous solution is by substituting the $y$y-value from the first equation into the second. Thus, $x+(x+1.3)=3.7$x+(x+1.3)=3.7. This simplifies to $2x=2.4$2x=2.4 and then, $x=1.2$x=1.2.
Finally, we find $y$y by substituting for $x$x in either one of the original equations. Thus, $y=1.2+1.3$y=1.2+1.3 and the solution can be written as $(x,y)=(1.2,2.5)$(x,y)=(1.2,2.5).
Having found a solution, we should check that it does indeed satisfy both equations.
Example 2
We wish to find the intersection of the lines $x-3y=-1$x−3y=−1 and $3x-y=13$3x−y=13.
This example has a simple whole-number solution and it may be possible to find it by trying out various possibilities. Other cases are not so simple. So, we show a method that always works.
If both sides of an equation are multiplied by the same number, the result must also be a true statement. In this case, we could multiply the first equation by $3$3 to get $3x-9y=-3$3x−9y=−3. This was done to make the $x$x coefficients the same in the two equations.
Now, we can eliminate the x-term by subtracting the left and right sides of this new equation from the second equation. That is
$3x-y-(3x-9y)=13-(-3)$3x−y−(3x−9y)=13−(−3)
This simplifies to $8y=16$8y=16 and hence, $y=2$y=2.
We can now substitute this $y$y-value back into one of the original equations to find the $x$x-value. If $x-3y=-1$x−3y=−1, then $x-3\times2=-1$x−3×2=−1 and so, $x=5$x=5.
This method is particularly suited to implementation in computer programs for solving much more complicated systems of linear equations.
Example 3
Consider the equation $y=-2x$y=−2x. A table of values is given below.
Plot the points in the table of values.
| $x$x | $-4$−4 | $-3$−3 | $-2$−2 | $-1$−1 |
|---|---|---|---|---|
| $y$y | $8$8 | $6$6 | $4$4 | $2$2 |
- Loading Graph...
Example 4
Consider the equations $y=-2x-8$y=−2x−8 and $y=5x+34$y=5x+34.
Fill in the $y$y-values for each of the $x$x-values given in the table:
$x$x $y=-2x-8$y=−2x−8 $y=5x+34$y=5x+34 $-7$−7 $\editable{}$ $\editable{}$ $-6$−6 $\editable{}$ $\editable{}$ $-5$−5 $\editable{}$ $\editable{}$ $-4$−4 $\editable{}$ $\editable{}$ Hence find the values for $x$x and $y$y which satisfy both the equations $y=-2x-8$y=−2x−8 and $y=5x+34$y=5x+34.
$x=\editable{},y=\editable{}$x=,y=
Example 5
Solve the following system of equations.
| $2x$2x | $-$− | $y$y | $=$= | $3$3 |
| $-4x$−4x | $+$+ | $2y$2y | $=$= | $-6$−6 |
Is it an inconsistent system or does it have infinitely many solutions?
It is an inconsistent system.
AIt has infinitely many solutions.
BThe solutions of the system can be written in the form of a set of ordered pairs $\left(x,y\right)$(x,y), where $x$x is any real number and $y$y is an expression in terms of $x$x.
Find an expression for $y$y in terms of $x$x. Give your final answer in the form '$y=\text{. . .}$y=. . .'