Generally speaking, those problems come up when there are two unknown or variables to solve.

Moreover, a system of equations is a set of two or more equations that must be solved at the same time

For this reason, a system could also be called simultaneous equations. The word simultaneous means "occuring at the same time"

I will only provide you with real life examples that lead to a system of linear equations and how to set up the system.

Then, you will be ready to solve a system of equations using one of the three methods.

The sum of two numbers is twenty and their difference is ten. What are the two numbers?

Here is how to set up the system:

Let x be the first number

Let y be the second number

Then,

x + y = 20

x − y = 10

You can also write:

x + y = 20

y − x = 10

You will get the same answers except that the values for x and y will be swapped

You have 24 coins in your pockets that are worth 4.50 dollars. How many coins are quarters? How many coins are dimes?

Here is how to set it up:

Let q be the number of quarters

Let d be the number of dimes

Then,

q + d = 24

25 × q + 10 × d = 450

The second equation is tricky. How did we get it?

Since 1 quarter equal to 25 cents, q quarter equal to 25 × q

If you had 6 quarters and you wanted to know how many cents are there for the 6 quarters, would you not do 6 × 25?

Just say to yourself that now instead of 6 quarters you have q quarters. Does that make sense?

In a similar way, since 1 dime equal to 10 cents, d dimes equal 10 × d

What about the 450? 4.50 dollars times 100 = 450 cents

Finally, since 25 × q represents how many cents you have for quarters and 10 × d represents how many cents you have for dimes, adding them should equal to the total of 450 cents

A cell phone plan offers 300 free minutes for a flat fee of 20 dollars. If your usage exceed 300 minutes, you pay 50 cents for each minute.

A second cell phone plan offers 500 free minutes for a flat fee of 30 dollars. If your usage exceed 400 minutes, you pay 30 cents for each minute

Model the cost of both plan with a system.

Here is how to set up the system:

Let x be the number of minutes you talk

Let y be the cost

y = 20 + 0.50 x

y = 30 + 0.30 x

Minus 0.50x from both sides in the first equation . Minus 0.30 x from both sides in the second equation

We get:

y − 0.50 x = 20

y − 0.30 x = 30

Type the problem and show us how you solve it!

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**solving system of linear equations by substitution**

y=2x
x+y=21
Replace y = 2x into the second equation.
we get:
x + 2x = 21
we get
3x = 21
divide both sides of the equation by 3
3x/3 …

**ALGEBRA II **

FIND THE VALUE OF K SO THAT THE TWO GIVEN LINES WILL BE PARALLEL.
3X-KY=7
2X+5Y=-1
First, put the two equations in standard form
3X - KY = 7
…

**cost of cd**

Andrew had $93.00. With this money, he could buy 3 CDs and 4 books. However, he bought only 2 CDs and 3 books and had $27.00 left. What was the cost …

**age problem**

The sum of Mira and Shaheena is 40 years. After 10 years Mia will be twice as old as Shaheena. What is Mia's age. What is Shaheena's age?
Solution …

**Length and Width**

The perimeter of a rectangle is 138 inches. The length exceeds the width by 65 inches. Find the length and width.
P = 2l + 2w
l = w + 65
138 = …

**college student**

Alvin's age is three times Elga's age. The sum of their ages is " 40 ". What is Elga's age?
Let x be Alvin's age
Let y be Elga's age
x + y = …

**rectangle word problem**

the length of a rectangle is 5 more than the width and the area is 24. Find the length and the width.
The formula of the area of a rectangle is
…

**Story word Problem: linear equation**

A salesperson earns a $200 per week in base pay plus one-twentieth of his/her dollar sales volume. Find the salesperson’s dollar sales volume if he/she …

**find each side of a triangle** Not rated yet

The perimeter of a triangle is 58 centimeters. The longest side is 2 centimeters less than the sum of the other two sides. Twice the shortest is 12 centimeters …

Now go to the lessons below to learn how to solve a system of linear equations. I have also included a system of linear equations solver.

System of linear equations solver