GOSEA 5 - Solving One-Step Equations

GOSA 5

Solving One-Step Equations

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By continuing to the next slide and completing this lesson, you testify that you are

the student who is enrolled in GOSEA at BCHS

with Mrs. Haley and that you are doing the lesson for yourself and no one else.

Before getting to the heart of this lesson, there are a few properties about equations you need to know.

Pay attention!

Take notes if necessary.

• Addition Property of Equality

• Subtraction Property of Equality

• Multiplication Property of Equality

• Division Property of Equality

Before we get to the properties, there is some vocabulary you must know.

• What is an equation?

• What is the difference between an expression and an equation?

• What is a solution to an equation?

• How many sides does an equation have?

What is an equation?

An equation is a mathematical statement that contains an equals sign (=).

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An equation is TWO (and only two) expressions joined

by AN (implying one) equals sign.

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Equations may be true, false or conditional.

What is the difference between an expression and an equation?

The word "expression" was defined in a previous lesson.

An "equation" is made up of expressions.

Expressions do not contain equals signs.

Equations contain ONE equals sign and, therefore, can only join two expressions. (Some teachers use and allow more than one equals sign. Mrs. Haley does not.)

True? False? Conditional?

True equations are true mathematical statements that contain an equals sign.

False equations are false mathematical statements that contain an equals sign.

Conditional equations are mathematical statements that contain an equals sign and that could be true or false depending on the value assigned to the variable.

What does it mean to "solve" an equation?

Only conditional equations can be "solved".

To solve an equation means to find the value of the variable that makes the conditional equation be a true equation.

The value that makes a conditional equation be true is called the "solution" to the equation.

How many sides does an equation have?

Equations contain ONE equals sign.

Because there is only ONE equals sign, there are only TWO sides of an equation.

I cannot stress enough the importance of understanding that there are only TWO SIDES to an equation.

I also apologize for any teacher that has made you think it is okay to have more than one equals sign in an equation.

In case you have not already figured it out, different teachers have different requirements. The same is true for math teachers.

Multiple Choice

True? False? Conditional?

3 + 4 = 7

• True

• False

• Conditional

Multiple Choice

True? False? Conditional? 3 $$\cdot$$ 4 = 7

• True

• False

• Conditional

Multiple Choice

True? False? Conditional?

3x = 12

• True

• False

• Conditional

Solution to an equation

The solution to an equation is the value of the variable that makes the conditional equation be a true equation.

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In the conditional equation 3x = 12, 4 is the value that when multiplied by 3 equals 12. Therefore, x = 4 is the solution to the equation.

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When writing answers to equations, do not write just "4".

The answer is "x = 4".​ This is important because some equations have more than one variable, and each variable will have its own value.

In order to solve a conditional equation, we must remember there only two sides to an equation.

(Now on to the properties.)

Addition (Additive) Property of Equality

• This property says that the same number can be added to both SIDES of an equation without changing the "truthness" (or solution) to the equation.

• Performing the same property to both sides of an equation keeps the equation "balanced".

• 3 + 4 = 7 is a true equation

• 3 + 4 + 10 = 7 + 10 is still true (and balanced) because the same number (10) was added to both sides of the equation.

Addition (Additive) Property of Equality (continued)

x + 2 = 5 is a conditional equation

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The solution to the above equation is x = 3 because 3 + 2 = 5.

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x + 2 + 8 = 5 + 8 has the same solution x = 3 because the same number

(8) was added to BOTH sides of the equation to keep it balanced.

Subtraction Property of Equality

• This property says the same number can be subtracted from both sides of an equation without changing the "truthness" (or solution) to the equation.

• Performing the same property to both sides of an equation keeps the equation "balanced".

• 8 + 7 = 15 is a true equation

• 8 + 7 - 5 = 15 - 5 is still true (and balanced) because the same number (5) was subtracted from both sides of the equation.

Subtraction Property of Equality (continued)

x + 12 = 20 is a conditional equation

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The solution to the above equation is x = 8 because 8 + 12 = 20.

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x + 12 - 6 = 20 - 6 has the same solution x = 8 because the same number

(8) was added to BOTH sides of the equation to keep it balanced.

How does one "solve" an equation?

• While the purpose of solving an equation is to find the value of the variable that makes the conditional equation become a true statement, the act of solving an equation is done by applying Properties of Equality.

• Applying the properties of equality allows us to keep the equation balanced and to "isolate" the variable so that we may find its value.

• To isolate a variable, we simply "undo" the operation(s) that is (are) indicated in the equation.

Solve: x - 10 = 20

• I know you can figure out the answer in your head, but you will have to show your work in Geometry.

• Knowing how to show your work requires that you understand how the properties work.

• To isolate x, you must undo the subtraction of 10 by adding 10 to the left side of the equation.

• To keep the equation balanced, you must apply the Addition Property of Equality and add 10 to the right side of the equation also.

• x - 10 + 10 = 20 + 10

  x + 0 = 0​

• x = 30 is the solution to the equation

Solve: y + 8 = 15

• To isolate y, you must undo the addition of 8 by subtracting 8 from the left side of the equation.

• To keep the equation balanced, you must apply the Addition Property of Equality andsubtract 8 from the right side of the equation also.

• y + 8 - 8 = 15 - 8

  y + 0 = 7​

• y = 7 is the solution to the equation

Summary

If the variable in an equation has addition acting on it, you will subtract to isolate the variable.

If the variable in an equation has subtraction acting on it, you will add to isolate the variable.

Multiple Choice

What operation will you use to isolate the variable?

h + 6 = 8

• addition

• subtraction

Multiple Choice

What operation will you use to isolate the variable?

m - 3 = 11

• addition

• subtraction

Multiple Choice

What operation will you use to isolate the variable?

9 + t = 14

• addition

• subtraction

Multiple Choice

What operation will you use to isolate the variable?

-5 + k = 8

• addition

• subtraction

Multiple Choice

What operation will you use to isolate the variable?

16 = b - 5

• addition

• subtraction

Multiple Choice

What operation will you use to isolate the variable?

-8 = b + 6

• addition

• subtraction

Multiple Choice

What is the correct step to solve x + 7 = 6 ?

• x + 7 + 7 = 6 + 7

• x + 7 - 7 = 6 - 7

• x + 7 - 7 = 6 + 7

• x + 7 - 6 = 6 - 6

Multiple Choice

What is the correct step to solve 10 = y - 5 ?

• 10 + 5 = y - 5 + 5

• 10 - 10 = y - 5 - 10

• 10 - 5 = y - 5 - 5

• 10 - 5 = y - 5 + 5

Multiplication (Multiplicative)

Property of Equality

• This property says that the same number can be multiplied by EVERY term on both SIDES of an equation without changing the "truthness" (or solution) to the equation.

• 3 + 4 = 7 is a true equation

• 3(5) + 4(5) = 7(5) is still true (and balanced) because the same number (5) was multiplied by every term on both sides of the equation.

Multiplication (Multiplicative)

Property of Equality (contiued)

• 3x = 15 is a conditional equation

• The solution to the equation is x = 5 becasue (3)(5) = 15

• 3x (2) = 15(2) has the same solution x = 5 because the same number (2) was multiplied by BOTH sides of the equation to keep it balanced.

• 3x (2) = 15(2)

  6x = 30

  x = 5 because (6)(5) = 30.

Division Property of Equality

Summary

If the variable in an equation has multiplication acting on it, you will divide to isolate the variable.

If the variable in an equation has division acting on it, you will multiply to isolate the variable.

Solve: 4y = 20

Multiple Choice

What operation will you use to isolate the variable?

5t = 40

• mutliplication

• division

Multiple Choice

What operation will you use to isolate the variable? $$\frac{h}{3}=\ 7$$  

• mutliplication

• division

Multiple Choice

What is the correct step for solving the equation?

10k = 120

• $$10k\ \cdot10\ =\ 120\ \cdot10$$

• $$\frac{10k}{10}=\ \frac{120}{10}$$

• $$\frac{10k}{120}=\ \frac{120}{120}$$

Multiple Choice

What is the correct step for the solving the equation?

$$\frac{d}{9}=\ 8$$  

• $$8\cdot\frac{d}{9}=8\cdot8$$  

• $$\frac{d}{9}\div9\ =\ 8\ \div9$$  

• $$9\cdot\frac{d}{9}=8\cdot9$$  

Reminder

If a negative number is being multiplied by the variable, you will divide by the negative number to isolate the variable.

If a variable is divided by a negative number, you will multiply by the negative number to isolate the variable.

Poll

How much assistance did you have to have to complete this assignment.

• Some assistance was needed

• No assistance was needed

​Keep doing this assignment until you make a 100. That is the only way you will get the 100 for Geometry this Fall.