Lesson: Chapter - 5
Systems of Equations
Sometimes, a question will have a lone equation containing two variables, and
using the methods we’ve discussed up until now will not be enough to solve for
the variables. Additional information is needed, and it must come in the form of
another equation.
Say, for example, that a single equation uses the two variables x and
y. Try as you might, you won’t be able to solve for x or y.
But given another equation with the same two variables x and y,
then the values of both variables can be found.
These multiple equations containing the same variables are called systems of
equations. For the Math IC, there are essentially two types of systems of
equations that you will need to be able to solve. The first, easier type
involves substitution, and the second involves manipulating equations
simultaneously.
Video Lesson - Substitution Method
Substitution
Simply put, substitution is when the value of one variable is found and then
substituted into the other equation to solve for the other variable. It can be
as easy as this example:
If
x – 4 =
y – 3 and 2
y = 6, what is
x?
In this case, we have two equations. The first equation contains x and
y. The second contains only y. To solve for x, you must solve
for y in the second equation and substitute that value for y in
the first equation. If 2y = 6, then y = 3, and then x =
y – 3 + 4 = 3 – 3 + 4 = 4.
Here is a slightly more complicated example.
Suppose 3
x =
y + 5 and 2
y – 2= 12
k. Solve for
x
in terms of
k.div>
Again, you cannot solve for x in terms of k using just the first
equation. Instead, you must solve for y in terms of k in the
second equation, and then substitute that value in the first equation to solve
for x.
2
y - 2 = 12
k
2
y = 12
k + 2
y = 6
k + 1
Then substitute
y = 6
k + 1 into the equation 3
x =
y
+ 5.
3
x =
y + 5
3
x = (6
k + 1) + 5
3
x = 6
k + 1
x = 2
k + 2
Video Lesson - Elimination Method
Simultaneous Equations
Simultaneous equations refer to equations that can be added or subtracted from
each other in order to find a solution. Consider the following example:
Suppose 2
x + 3
y = 5 and –1
x – 3
y = –7. What is
x?
In this particular problem, you can find the value of x by adding the two
equations together:
Here is another example:
6
x + 2
y = 11 and 5
x + y = 10. What is
x + y?
By subtracting the second equation from the first:
Some test-takers might have seen this problem and been tempted to immediately
start trying to solve for x and y individually. The better
test-taker notices that by subtracting the second equation from the first, the
answer is given.
Give this last example a try:
2
x + 3
y = –6 and –4
x + 16
y = 13. What is the value
of
y?
The question asks you to solve for y, which means that you should find a
way to eliminate one of the variables by adding or subtracting the two
equations. 4x is simply twice 2x, so by multiplying the first
equation by 2, you can then add the equations together to find y.
2 ×(2
x + 3
y = –6) = 4
x + 6
y = –12
Now add the equations and solve for
y.
When you solve for one variable, like we have in this last example, you can
solve for the second variable using either of the original equations. If the
last question had asked you to calculate the value of xy, for example,
you could solve for y, as above, and then solve for x by
substitution into either equation. Once you know the independent values of x
and y, you can multiply them together.
Simultaneous equations on the Math IC will all be this simple. They will have
solutions that can be found easily by adding or subtracting the equations given.
Only as a last resort should you solve for one variable in terms of the other
and then plug that value into the other equation to solve for the second
variable.
Next to display next topic in the chapter.
Mathematics Practice Questions
Video Lessons and 10 Fully Explained Grand Tests
Large number of solved practice MCQ with explanations. Video Lessons and 10 Fully explained Grand/Full Tests.