A system of equations
is a group of two or more equations that
share variables. In order to solve a system
of equations, you need to have at least
one equation for every variable. So if there
are three variables in the system (x, y
and z), then we need to have at least three
equations in order to solve.
There are several different methods used
to solve a system of equations, including
substitution, elimination,
and addition.
Let’s start by looking at an example of
a system of equations:
3y=4x+3
y-4=3x+2
Substitution Method
To use the substitution method, you must
have a group of two or more equations that
share variables (as mentioned above). The
example appears to meet this condition,
let’s go over the steps needed to solve
using this method:
1. Choose an equation and isolate a variable
(usually the easiest variable to solve for).
2. Substitute the value of the variable
that you solved for into the second equation
and solve for the variable in that equation.
3. Substitute the value found in the second
step back into the first equation to solve
for the second variable.
4. Substitute the values found for both
variables into both equations to check your
work and prove that they are correct.
Now let’s go through each of the steps listed
above to solve the system of equations:
3y=4x+3
y-4=3x+2
1. Since the ‘y’ in the second equation
appears easiest to solve for, we’ll use
that as our starting point.
y-4=3x+2
y=3x+2+4
y=3x+6
2. Substitute the solution from the first
step into the other equation and solve for
the variable in that equation.
3y=4x+3
y=3x+6 (the value of y;
substitute this into the equation)
3(3x+6)=4x+3
9x+18=4x+3
5x+18=3
5x=-15
x=-3
3. Substitute the value found in the second
step (x=-3) into the first equation and
solve for the other variable.
3y=4x+3
3y=4(-3)+3
3y=-12+3
3y=-9
y=-3
So the solution to this system of equations
is x=-3, and y=-3.
4. To check your work, substitute the solution
into the other equation and solve.
y-4=3x+2
(-3)-4=3(-3)+2
-7=-9+2
-7=-7
Since -7=-7 is a true statement, the solution
is correct. |
