150r. Review Equations, Solutions and Inspection

Learning Intentions

• To understand that an equation is a mathematical statement that can be true or false
• To understand that a solution is a value for the unknown that makes an equation true
• find a solution to simple equations by inspection
• write equations from worded descriptions

Pre-requisite Summary

• Know that an expression represents a quantity, while an equation compares two quantities
• Understand that the equals sign means the values on both sides are the same
• Be able to substitute a value into an expression
• Know that a pronumeral or variable stands for an unknown number
• Be able to use basic number facts for addition, subtraction, multiplication and division
• Understand that checking means substituting a value back into the equation
• Be able to translate simple word phrases into algebraic expressions

Worked Examples

Worked Example 1

State whether each equation is true or false:
a) $3+4=7$
b) $9-2=10$
c) $2x+1=7$ when $x=3$

Worked Example 2

State the solution of each equation by inspection:
a) $x+5=12$
b) $m-3=8$
c) $2p=14$

Worked Example 3

State the solution of each equation by inspection:
a) ${\displaystyle \frac{q}{4}}=3$
b) $y+9=15$
c) $3a=21$

Worked Example 4

Check whether the given value is a solution:
a) Is $x=4$ a solution of $x+6=10$?
b) Is $n=5$ a solution of $2n=12$?
c) Is $b=3$ a solution of $b-1=2$?

Worked Example 5

Write an equation for each description:
a) A number $x$ increased by $4$ equals $11$
b) Three times a number $y$ is $18$
c) A number $p$ divided by $5$ is $7$

Worked Example 6

Write an equation from each worded situation, then solve by inspection:
a) Sam has $x$ marbles. After getting $3$ more, he has $10$ marbles
b) A packet has $m$ stickers. After using $4$, there are $9$ left
c) A chair costs $\mathrm{\$}q$ each. Two chairs cost $\mathrm{\$}30$

Problems

Problem 1

State whether each equation is true or false:
a) $6+2=8$
b) $12-5=9$
c) $3x=15$ when $x=5$

Problem 2

State the solution of each equation by inspection:
a) $x+7=13$
b) $m-4=6$
c) $2p=18$

Problem 3

State the solution of each equation by inspection:
a) ${\displaystyle \frac{q}{3}}=4$
b) $y+8=14$
c) $4a=28$

Problem 4

Check whether the given value is a solution:
a) Is $x=5$ a solution of $x+4=9$?
b) Is $n=4$ a solution of $3n=15$?
c) Is $b=6$ a solution of $b-2=4$?

Problem 5

Write an equation for each description:
a) A number $x$ increased by $6$ equals $14$
b) Four times a number $y$ is $20$
c) A number $p$ divided by $6$ is $5$

Problem 6

Write an equation from each worded situation, then solve by inspection:
a) Mia has $x$ books. After buying $2$ more, she has $9$ books
b) A jar has $m$ lollies. After eating $5$, there are $7$ left
c) A pen costs $\mathrm{\$}q$ each. Three pens cost $\mathrm{\$}21$

Exercises

Understanding and Fluency

1. State whether each statement is an equation or not:
   a) $x+3$
   b) $x+3=8$
   c) $5=2+3$
   d) $4y$

2. State whether each equation is true or false:
   a) $8+1=9$
   b) $14-6=7$
   c) $2\times 5=10$
   d) $12÷3=5$

3. Find the solution by inspection:
   a) $x+4=9$
   b) $y-2=7$
   c) $3m=12$
   d) ${\displaystyle \frac{p}{2}}=6$

4. Find the solution by inspection:
   a) $a+10=18$
   b) $b-5=11$
   c) $4c=24$
   d) ${\displaystyle \frac{d}{5}}=3$

5. Check whether the given value is a solution:
   a) Is $x=6$ a solution of $x+2=8$?
   b) Is $y=3$ a solution of $2y=8$?
   c) Is $m=7$ a solution of $m-4=3$?
   d) Is $p=10$ a solution of ${\displaystyle \frac{p}{5}}=2$?

6. Write an equation for each description:
   a) A number $x$ plus $5$ equals $13$
   b) A number $y$ minus $7$ equals $4$
   c) Twice a number $m$ is $16$
   d) A number $p$ divided by $4$ is $6$

7. Write an equation from each situation:
   a) A bag has $b$ apples. After adding $6$, there are $15$ apples
   b) A score is $s$. After losing $3$ points, the score is $11$
   c) Five equal boxes hold $20$ pencils in total
   d) A ribbon of length $r$ cm is cut into $3$ equal pieces of $4$ cm each

8. Write the equation, then solve by inspection:
   a) A number increased by $8$ is $17$
   b) Three times a number is $27$
   c) A number divided by $7$ is $6$
   d) A number decreased by $9$ is $5$

Reasoning

9. Explain why an equation can be true or false.

10. A student says that $x+5$ is an equation because it has a variable. Explain the mistake.

11. Noah says that if $x=4$ makes one side of the equation equal to $9$, then it is automatically a solution. Is he correct? Explain.

12. Explain why a solution must make both sides of an equation equal.

13. A student says that the solution to $2x=10$ is $x=8$ because $10-2=8$. Describe the error.

Problem-solving

14. A game score starts at $x$ points. After gaining $6$ points, the score is $14$. Write an equation and find $x$.

15. A pack contains $m$ markers. After giving away $4$, there are $9$ left. Write an equation and find $m$.

16. Four equal bags contain $28$ marbles altogether. Write an equation and find how many marbles are in one bag.

17. A movie ticket costs $\mathrm{\$}p$. Two tickets cost $\mathrm{\$}24$. Write an equation and find $p$.

18. A rope of length $r$ m is cut so that $5$ m remains after $3$ m is removed. Write an equation and find $r$.

19. A student has $x$ stickers. After doubling the number, the total is $18$. Write an equation and find $x$.

Potential Misunderstandings

• Students may think any algebraic expression is an equation
• Students may forget that an equation must include an equals sign
• Students may think a solution is any number that can be substituted, rather than one that makes the equation true
• Students may check only one side of the equation instead of both sides
• Students may confuse solving by inspection with using a random guess without checking
• Students may reverse word phrases such as “a number increased by $4$ equals $11$”
• Students may write an expression instead of an equation from a worded description
• Students may use the wrong operation when finding a solution by inspection