A system of equations 3×3 is a system consisting of three equations with three variables. There are several methods to solve a system of equations 3×3, such as graphing, substitution, and Gaussian elimination.
In this article, we will learn how to solve systems of equations 3×3 using the substitution and elimination methods. We will look at several solved exercises to learn about this topic.
How to solve systems of equations 3×3?
To solve a system of three equations with three variables, we can use one of several methods. Here are two common approaches:
- Gaussian elimination: This method consists of adding or subtracting equations to eliminate variables, one at a time until the system is in what is known as row-echelon reduced form. Once the system is in this form, it is easy to solve for the variables.
- Cramer’s rule: This method consists of expressing the solution in terms of determinants of certain matrices. To use Cramer’s rule, it is necessary to be able to calculate determinants, which can be a little more complicated than the methods used in Gaussian elimination.
Here is an example of how to solve a system of three equations with three variables using Gaussian elimination:
Suppose we have the following system:
$$\begin{cases} 3x + 4y – 2z = 0\\2x – 3y + 4z = 11\\x – 2y + 3z = 7 \end{cases}$$
We can start by eliminating the variable x from the second and third equations. To do this, we multiply the second equation by 1 and the third equation by -2, and then add the resulting equations:
$$\begin{cases} 2x – 3y + 4z = 11\\-2x + 4y -6z = -14 \end{cases}$$
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$latex y-2z=-3$
Then, we can eliminate the variable x from the first and third equations by multiplying the first equation by 1 and the third equation by -3, and then adding the resulting equations:
$$\begin{cases} 3x+4y-2z=0\\-3x+6y-9z=-21 \end{cases}$$
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$latex 10y-11z=-21$
Finally, we can multiply the equation $latex y-2z=-3$ by -10 and add these two equations to eliminate the y variable:
$$\begin{cases} -10y+20z=30\\10y-11z=-21 \end{cases}$$
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$latex 9z=9$
Solving this equation for z, we find that $latex z = 1$.
Substituting this value into the equation $latex y-2z=-3$, we can solve for y: $latex y = -1$.
Substituting these values back into the original equations, we can solve for x: $latex x = 2$.
Therefore, the solution for the system is $latex x = 2$, $latex y = -1$ and $latex z = 1$.
Systems of equations 3×3 – Examples with answers
EXAMPLE 1
Find the solution to the following system of three equations with three unknowns by the substitution method:
$$\begin{cases} x+y-2z=1\\2x-4y+z=0\\2y-3z=-1 \end{cases}$$
Solution
We number the equations as (i), (ii), (iii):
$$\begin{cases} \text{(i)}~~ x+y-2z=1\\\text{(ii)}~~ 2x-4y+z=0\\ \text{(iii)}~~2y-3z=-1 \end{cases}$$
From the third (iii), we isolate z:
$latex z=\frac{2}{3}y+\frac{1}{3} $
Substitute z in (ii) to obtain (ii’):
$$2x-4y+\frac{2}{3}y+\frac{1}{3}=\ 2x-\frac{10}{3}y+\frac{1}{3}=0 \; \; \text{(ii’)}$$
We also substitute z in (i) to obtain (i’):
$$ x+y-2(\frac{2}{3}y+\frac{1}{3})=\, x-\frac{1}{3}y-\frac{2}{3}=1 \; \; \text{(i’)} $$
From the above equation (i’) we isolate y: $latex y=3x-5 $.
Substituting y in (ii’), we get: $latex 17-8x=0$. Subtracting, we are left with:
$$ x=\frac{17}{8}$$
Substitute the found value of x in (i’) and isolate y: $latex \frac{17}{8}-\frac{1}{3}y-\frac{2}{3}=1$
to obtain:
$$y=\frac{11}{8}$$
Finally, we substitute x and y in the equation (i) and solve for z:
$$\frac{17}{8}+\frac{11}{8}-2z=1$$
$$z=\frac{5}{4}$$
Summarizing, the solution to the system is:
$$\left[ x=\frac{17}{8},y=\frac{11}{8},z=\frac{5}{4}\right] $$
EXAMPLE 2
We have the following system of equations of three linear equations with three unknowns:
$$\begin{cases} \text{(i)}~~x+y+z=6 \\ \text{(ii)}~~x-y+2z=5 \\ \text{(iii)}~~x-y-3z=-10 \end{cases}$$
Show, by the elimination method that the solution is: $latex \left[ x=1,y=2,z=3 \right] $.
Solution
Subtracting (i) minus (ii) we obtain: $latex 2y-z=1 $
Subtracting (ii) minus (iii) gives: $latex 5z=15$.
It follows that: $latex z=3 $
Substituting in the penultimate expression, we are left with: $latex 2y-3=1 $, whose solution is: $latex y=2 $.
Finally, we substitute the values obtained in equation (i): $latex x+2+3=6$.
And we get: $latex x=1 $.
EXAMPLE 3
Having the following system of equations
$$\begin{cases} \text{(i)}~~2x+3y+z=1 \\ \text{(ii)}~~6x-2y-z=-14 \\ \text{(iii)}~~3x+y-z=1 \end{cases}$$
Find the values of x, y, z that satisfy the system.
Solution
We will apply the elimination method.
Multiply equation (i) by 3 and equation (ii) by -1 and add them together:
$$3(2x+3y+z)-(6x-2y-z)=3(1)-(-14) $$
x is canceled to obtain: $latex 11y+4z=17 $ (I)
Multiply equation (ii) by 1 and equation (iii) by -2 and add them together:
$$ (6x-2y-z)-2(3x+y-z)=(-14)-2(1) $$
x is eliminated to obtain: $latex z-4y=-16 $ (II)
Subtract (I) minus 4(II):
$$(11y+4z)-4(z-4y)=(17)-4(-16)$$
z is eliminated to have: $latex 27y=81 $ (III)
From eq.(III) we solve for y: $latex 27y=81 $, and we obtain: $latex y=3 $.
Substitute the obtained value of y in (II): $latex z-4(3)=-16 $, whose solution is: $latex z=-4 $.
Finally, we substitute the values obtained for z and y in (i): $latex 2x+3(3)+(-4)=1$.
By subtracting x: $latex 2x+3(3)+(-4)=1$, we obtain: $latex x=-2$.
The solution to the system is: $latex \left[ x=-2,y=3,z=-4\right] $
EXAMPLE 4
We have the following system of three equations with three unknowns:
$$\begin{cases} \text{(i)}~~\frac{x}{2}+\frac{y}{2}-\frac{z}{3}=3 \\ \text{(ii)}~~ \frac{x}{3}+\frac{y}{6}-\frac{z}{2}=-5 \\ \text{(iii)}~~\frac{x}{6}-\frac{y}{3}+\frac{z}{6}=0 \end{cases}$$
Obtain the solution by the substitution method.
Solution
From (iii) we solve for z: $latex \frac{x}{6}-\frac{y}{3}+\frac{z}{6}=0 $,
And we get: $latex z=2y-x $ (iii’)
Substituting in (ii): $latex \frac{x}{3}+\frac{y}{6}-\frac{2y-x}{2}=-5 $
Now, we isolate y: $latex y=x+6 $ (ii’)
Substituting (ii’) in (iii’): $latex z=2(x+6)-x=\; x+12 $, that is: $latex z=x+12 $ (iv)
The values obtained in (ii’) and (iv) are used in equation (i):
$$\frac{x}{2}+\frac{x+6}{2}-\frac{x+12}{3}=3 $$
And the variable x is isolated: $latex x=6$.
Now substitute the value obtained in (ii’) and solve for y: $latex y=6+6=12$.
Finally, we substitute the value of x in (iv): $latex z=6+12=18$.
In summary, the solution to the system of equations is: $latex \left[ x=6,y=12,z=18 \right] $
EXAMPLE 5
Find the solutions to the following system of equations:
$$\begin{cases} \text{(i)}~~x+y=5 \\ \text{(ii)}~~ x+z=6 \\ \text{(iii)}~~ y+z=7 \end{cases}$$
Solution
Subtracting (i) minus (ii) we obtain:
$latex y-z=-1$ (ii’)
Subtracting the above equation (ii’) minus (iii) gives:
$$-2z=-8$$
It follows that: $latex z=4$.
Substituting the value obtained for z in eq.(iii) we are left with:
$latex y+4=7$, from which we obtain: $latex y=3$.
Finally, the obtained value of y is substituted in (i):
$latex x+3=5$, from which we obtain: $latex x=2$.
In summary, the system of equations has the following solution:
$$\left[ x=2,y=3,z=4\right] $$
EXAMPLE 6
Solve the following system of equations:
$$\begin{cases} \text{(i)}~~3x+2y=2 \\ \text{(ii)}~~ 2y+2z=\frac{3}{2} \\ \text{(iii)}~~x+4z=\frac{4}{3} \end{cases}$$
Solution
We will apply the elimination method.
Subtracting equation (ii) from (i):
$$(3x+2y=2)-\left(2y+2z= \frac{3}{2} \right)$$
==> $latex 3x-2z = \dfrac{1}{2}$ (I)
2(I) + (iii):
$$2(3x-2z=\frac{1}{2})+\left(x+4z=\frac{4}{3}\right)$$
==> $latex 7x=\frac{7}{3}$
==> $latex x=\frac{1}{3}$
Substituting $latex x=\frac{1}{3}$ in (i):
$latex 3(\frac{1}{3})+2y=2$ ==> $latex 2y+1=2$,
getting: $latex y=\frac{1}{2}$
The obtained value of y is substituted in (ii): $latex 2(\frac{1}{2})+2z=3/2$
and we get: $latex z=\frac{1}{4}$
Summarizing, the solution to the system of equations is:
$$\left[ x=\frac{1}{3},y=\frac{1}{2},z=\frac{1}{4}\right] $$
EXAMPLE 7
Find the values of the variables u, v, and w in the following system of equations:
$$\begin{cases} \text{(i)}~~\frac{1}{u}+\frac{1}{v}=5 \\ \text{(ii)}~~ \frac{1}{u}+\frac{1}{w}=6 \\ \text{(iii)}~~\frac{1}{v}+\frac{1}{w}=7 \end{cases}$$
Solution
We start by making the following variable change:
$$x=\frac{1}{u}$$
$$y=\frac{1}{v}$$
$$z=\frac{1}{w}$$
To obtain:
$$\begin{cases} \text{(i)}~~x+y=5 \\ \text{(ii)}~~ x+z=6 \\ \text{(iii)}~~y+z=7 \end{cases}$$
The solution of this system is: $latex \left[ x=2,y=3,z=4\right] $ (see example 5).
The change of variable previously made is now reversed:
$latex u=\frac{1}{x}$ sustituyendo el x=2 se obtiene: $latex u=\frac{1}{2}$
$latex v=\frac{1}{y}$ sustituyendo el y=3 se obtiene: $latex v=\frac{1}{3}$
$latex w=\frac{1}{z}$ sustituyendo el z=4 se obtiene: $latex w=\frac{1}{4}$
EXAMPLE 8
Solve the following system of equations:
$$\begin{cases} \text{(i)}~~ \frac{3}{u}+\frac{2}{v}=2 \\ \text{(ii)}~~\frac{2}{v}+\frac{2}{w}=\frac{3}{2} \\ \text{(iii)}~~ \frac{1}{u}+\frac{4}{w}=\frac{4}{3} \end{cases}$$
Solution
The following variable change is made:
$$ x=\frac{1}{u}$$
$$y=\frac{1}{v}$$
$$z=\frac{1}{w}$$
To obtain:
$$\begin{cases} \text{(i)}~~ 3x+2y=2 \\ \text{(ii)}~~2y+2z=\frac{3}{2} \\ \text{(iii)}~~ x+4z=\frac{4}{3} \end{cases}$$
and its solution is: $latex \left[ x=\frac{1}{3},y=\frac{1}{2},z=\frac{1}{4}\right] $ (see example 6)
The change of variable previously made is now reversed:
$latex u=\frac{1}{x}$ substituting x=1/3 we get: $latex u=3$
$latex v=\frac{1}{y}$ substituting y=1/2 we get: $latex v=2$
$latex w=\frac{1}{z}$ substituting z=1/4 we get: $latex w=4$
EXAMPLE 9
Find the values of x, y, and z that simultaneously satisfy the three given equations:
$$\begin{cases} \text{(i)}~~x+y=1 \\ \text{(ii)}~~y+z=-1 \\ \text{(iii)}~~x+z=-6 \end{cases}$$
Solution
Subtracting (i) minus (ii) :
$latex (x+y=1)-(y+z=-1)$ we get: $latex x-z=2$ (ii’)
Adding the above equation (ii’) plus (iii) :
$latex(x-z=2)+(x+z=-6)$ we get: $latex 2x=-4$
It follows that: $latex x=-2$.
Substituting the value obtained for x in eq.(i), we are left with:
$latex -2+y=1$, we get: $latex y=3$
Finally, the obtained value of y is substituted in (ii):
$latex 3+z=-1$, we get: $latex z=-4$
In summary, the system of equations has the following solution:
$$\left[ x=-2,y=3,z=-4\right] $$
EXAMPLE 10
Solve the following system of equations:
$$\begin{cases} \text{(i)}~~ 5x-3z=2 \\ \text{(ii)}~~ -y+2z=-5 \\ \text{(iii)}~~ x+2z=8 \end{cases}$$
Solution
We will apply the elimination method.
2(i) + 3(ii):
$$2(5x-3z=2)+3(-y+2z=-5)$$
==> $latex 10x-3y=-11$ (I)
(ii) – (iii):
$$(-y+2z=-5)-(x+2z=8)$$
==> $latex x+y=13$ (II)
(I)+3(II):
$$(10x-3y=-11)+3(x+y=13)$$
which reduces to: $latex13x=28$, obtaining: $latex x=\frac{28}{13}$
Substituting $latex x=\frac{28}{13}$ in (iii): $latex \frac{28}{13}+2z=8$, we find the value of z: $latex z=\frac{38}{13}$
The obtained value of z is substituted in (ii): $latex -y+2(\frac{38}{13})=-5$, and we are left with: $latex y=\frac{141}{13}$
Summarizing, the solution to the system of equations is:
$$\left[ x=\frac{28}{13},y=\frac{141}{13},z=\frac{38}{13}\right]$$
Systems of equations 3×3 – Practice problems
Find the value of x in the system of equations: $$\begin{cases} 2x-2y+z=-5 \\ 3x+y+3z=-1 \\ 4x-y-2z=-12 \end{cases}$$
Write the value of x in the input box.
See also
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Jefferson Huera Guzman
Jefferson is the lead author and administrator of Neurochispas.com. The interactive Mathematics and Physics content that I have created has helped many students.
