#### Solving Systems of Equations - Substitution

y = 5x + 10

5x + 2y = 80

What does it mean to solve a system of linear equations?

What does it mean to solve by substitution?

The Mechanics of the Substitution Method

How do I know my solution is correct?

Another Consideration

Linear simply means a straight line, and a system in this case means more than one line. Above, there are equations for two lines. Curious minds want to know if these lines intersect. If so, where do they intersect? The point of intersection is considered the solution and designated by the (x, y) coordinates. In the real world, systems of linear equations are used many times to represent and solve situations concerning consumer economics as well as geometry.

In sports, substitution means one player will come out of the game and is replaced by another player. Likewise, in an equation, one of the players / variables, “x or “y” will be replaced. Pause. Remember, the “=” sign stands for the word “is”. Thus, the equation, y = 5x + 10, reads “y is 5x + 10,” or the value of y is “5x +10”. So, wherever you see the variable “y” in the other equations, you know that y comes out and is replaced with its value. After the substitution is made for “y,” you will notice there are no longer two variables but one. Thus, you are able to combine like terms and solve for the only variable left, “x”.

The original problem:

y = 5x + 10

5x + 2y = 80

5x + 2(5x + 10) = 80 ……… make substitution

5x + 2(5x) + 2(10) = 80 ……. use the distributive property

5x + 10x + 20 = 80 ………... multiply

15x + 20 = 80 ……………… combine like terms

15x + 20 – 20 = 80 – 20 …… subtract twenty from both sides

15x = 60 ……………………. Results of Subtraction

15x / 15 = 60 /15 …………… Divide both sides by 15

X = 4 ……………………….. Results of division

“X = 4” is the first coordinate of the solution.

Find the value of ”Y”

Choose one of the original equations above and make a substitution for “x.”

Y=5x +4

In part one, we determined x is 4.

So, y = 5(4) + 10

Y = 20 + 10

Y = 30 ……………. Is the second coordinate of the solution.

Solution = (X, Y) = (4, 30)

The solution implies that the values of x and y can be replaced in each equation, and the equation will be a true statement.

Therefore, let’s make substitution for both variables.

y =5x + 10 .........original equation

30 = 5(4) + 10 …. Substitute

30 = 20 + 10

30 = 30 True

5x + 2y = 80 ……….... original equation

5(4) + 2(30) = 80 ......... substitution

20 + 60 = 80

80 = 80 True

Thus, the solution (4, 30) is correct!

……………………

Sometimes, neither equation in the system has been solved for a variable. For example, our previous problem was

y = 5x + 10

5x + 2y = 80

The first equation was already solved for y. However, the problem could have been presented as

y - 5x = 10

5x + 2y =80

Then, you would have to solve for the “x” or “y” variable in one of the equations. Let’s choose

y – 5x = 10

y – 5x + 5x = 10 + 5x …….. add 5x to both sides

y = 5x + 10 ……….. result of addition

Now, you can continue with the mechanics of the substitution method.

5x + 2y = 80

What does it mean to solve a system of linear equations?

What does it mean to solve by substitution?

The Mechanics of the Substitution Method

How do I know my solution is correct?

Another Consideration

**What does it mean to solve a system of linear equations?**Linear simply means a straight line, and a system in this case means more than one line. Above, there are equations for two lines. Curious minds want to know if these lines intersect. If so, where do they intersect? The point of intersection is considered the solution and designated by the (x, y) coordinates. In the real world, systems of linear equations are used many times to represent and solve situations concerning consumer economics as well as geometry.

**What does it mean to solve by substitution?**In sports, substitution means one player will come out of the game and is replaced by another player. Likewise, in an equation, one of the players / variables, “x or “y” will be replaced. Pause. Remember, the “=” sign stands for the word “is”. Thus, the equation, y = 5x + 10, reads “y is 5x + 10,” or the value of y is “5x +10”. So, wherever you see the variable “y” in the other equations, you know that y comes out and is replaced with its value. After the substitution is made for “y,” you will notice there are no longer two variables but one. Thus, you are able to combine like terms and solve for the only variable left, “x”.

**The Mechanics of the Substitution Method**The original problem:

y = 5x + 10

5x + 2y = 80

**Part One:**5x + 2(5x + 10) = 80 ……… make substitution

5x + 2(5x) + 2(10) = 80 ……. use the distributive property

5x + 10x + 20 = 80 ………... multiply

15x + 20 = 80 ……………… combine like terms

15x + 20 – 20 = 80 – 20 …… subtract twenty from both sides

15x = 60 ……………………. Results of Subtraction

15x / 15 = 60 /15 …………… Divide both sides by 15

X = 4 ……………………….. Results of division

“X = 4” is the first coordinate of the solution.

**Part Two:**Find the value of ”Y”

Choose one of the original equations above and make a substitution for “x.”

Y=5x +4

In part one, we determined x is 4.

So, y = 5(4) + 10

Y = 20 + 10

Y = 30 ……………. Is the second coordinate of the solution.

Solution = (X, Y) = (4, 30)

**How do I know my solution is correct?**The solution implies that the values of x and y can be replaced in each equation, and the equation will be a true statement.

Therefore, let’s make substitution for both variables.

y =5x + 10 .........original equation

30 = 5(4) + 10 …. Substitute

30 = 20 + 10

30 = 30 True

5x + 2y = 80 ……….... original equation

5(4) + 2(30) = 80 ......... substitution

20 + 60 = 80

80 = 80 True

Thus, the solution (4, 30) is correct!

……………………

**Another consideration -**Sometimes, neither equation in the system has been solved for a variable. For example, our previous problem was

y = 5x + 10

5x + 2y = 80

The first equation was already solved for y. However, the problem could have been presented as

y - 5x = 10

5x + 2y =80

Then, you would have to solve for the “x” or “y” variable in one of the equations. Let’s choose

y – 5x = 10

y – 5x + 5x = 10 + 5x …….. add 5x to both sides

y = 5x + 10 ……….. result of addition

Now, you can continue with the mechanics of the substitution method.

**You Should Also Read:**

How to Find the Slope

Higher Math in the Workplace

Find the Equation of a Line

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