# How do I solve systems of equations and substitution word problems?

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### 3 Answers

Systems of equations can be solved to determine the value of the variables used in the equations. There are broadly two ways of solving a system of linear equations, by elimination and by substitution.

Let us take a simple example to illustrate how the two methods can be used for solving the same system of equations.

The system of equations to be solved is:

x - y = 7 ...(1)

x + y = 9 ...(2)

Adding (1) and (2) gives x - y + x + y = 7 + 9

=> 2x = 16

As you can see, the variable y has been eliminated and this allows the value of x to be determined.

=> x = 8

Similarly, (1) - (2) gives x - y - x - y = 7 - 9

=> -2y = -2

=> y = 1

The solution of the system of equations is (8, 1).

To solve, the system using substitution, write one of the equations such that it is possible to express one of the variables in terms of the other.

x - y = 7

=> x = y + 7

Now substitute this for x in (2)

x + y = 9

=> y + 7 + y = 9

=> 2y = 2

=> y = 1

Similarly from (1) we can get y = x - 7

Substituting this for x in (2)

x + x - 7 = 9

=> 2x = 16

=> x = 8

This gives the same solution as the earlier method.

There are majorly two ways of solving the systems of equations, preferably linear equations. They are substitution method or elimination method. Each of these has a distinct format of solution.

In order to solve substitution word problems, we use the following method, and to explain the method, let’s take up an example.

For example there are two equations:

y=x+1 and y=-2x-4

Now,

In order to find the value of x and y, we first substitute the value of any one variable from one equation by isolating the variable (say y in equation (i), and use it in the other to find out the value of one variable by putting it in equation (ii), then use that value derived (say, value of x) and put it in equation (i) to find out the value of the second variable( say, y).

Let,

y=x+1 ----(i)

y=-2x-4----(ii)

As given in the above explanation, we will now see if the variable y is already isolated or not in equation (i). Since it is already isolated, we take that value of y and insert it in place of y in the (ii) equation:

x+1 = 2x-4

1=2x-4-x

1+4=2x-x

2=x

or x=2

Now we put the value of x in equation (i) to derive the value of y:

y=x+1

y= 2+1

y=3

Therefore, the values of x and y have been derived by using the substitution method, which is merely based on substituting the value of one variable from one equation and using it in the other to derive the value of the other.

A system of equations can be solved in a couple different ways, but a common way is by substituting. When you are given a system of equations, you have two equations with the same variables that you have to solve for. An example would be

2x-3y=-2

4x+y=24

How you would solve this using substitution is to solve for one variable and put that answer into the other equation to get the second variable. For this example, I'll solve for y first with the second equation.

y=-4x+24

Put this into the y value for the other equation to solve for x.

2x-3(-4x+24)=-2

2x-(-12x+72)=-2

14x-72=-2

14x=70

x=5

Now you can put in this numerical value of x to get the numerical value of y (put it into any equation you want) and you get y=4.

For substitution into word problems, you are usually given an equation, or you'll be asked to create one, and you can simply put in the values it gives you. If the word problem says solve x+y when x=5 and y=4, you just put those numbvers into the probelm.

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