Expressing Relationships as Equations

1.6.2 Expressing Relationships as Equations

1.6.2 • Expressing Relationships as Equations

Activity

In questions 1 – 3, write as many equations as possible that could represent the relationship between the ages of the two children in each family described. Be prepared to explain what each part of your equation represents.

In Family A, the youngest child is 7 years younger than the oldest, who is 18.

Compare your answer:

Your answer may vary, but here are some samples. For $y$ , the age of the youngest child:

$y = 18 - 7$
$y + 7 = 18$
$18 - y = 7$

Where 11 is the age of the youngest child:

$18 - 7 = 11$
$11 + 7 = 18$
$18 - 11 = 7$

In Family B, the middle child is 5 years older than the youngest child.

Compare your answer:

Your answer may vary, but here are some samples. For $m$ , the age of the middle child, and $y$ , the age of the youngest

$m = y + 5$
$y = m - 5$

Tyler thinks that the relationship between the ages of the children in Family B can be described with $2 m - 2 y = 10$ , where $m$ is the age of the middle child and $y$ is the age of the youngest. Describe how Tyler came up with this equation.

Compare your answer:

Your answer may vary, but here ares some samples. The equation $m = y + 5$ , represents the relationship of the ages in Family B, where $m$ is the age of the middle child and $y$ the age of the youngest. Subtracting $y$ from both sides of the equation and multiplying both sides by 2 results in the equation: $2 m - 2 y = 10$ .

When the middle child is 12, the youngest child is 7, and that substituting $m = 12$ and $y = 7$ to $2 m - 2 y$ gives $2 (12) - 2 (7)$ or $24 - 14$ , which is 10. At all other ages of the two children, the expression $2 m - 2 y$ always has a value of 10.
$2 m - 2 y = 10$ is twice of $m - y = 5$ . If the difference between m and y is 5, then twice the difference between m and y must be twice of 5, which is 10, so the two equations are still describing the same relationship.

Select three equations that are equivalent to $3 a + 6 = 15$ .

$a + 3 = 12$
$3 a = 9$
$\frac{1}{3} a = 1$
$a + 2 = 5$

$\frac{1}{3} a = 1$ , $a + 2 = 5$ , and $3 a = 9$ .

Explain your reasoning for the equivalent equations in number 4.

Compare your answer:

Your answer may vary, but here are some samples.

The equation $3 a + 6 = 15$ can be divided by 3 on both sides and becomes $a + 2 = 5$ .
When 6 is subtracted from both sides of the equation $3 a + 6 = 15$ , the equation becomes 3a = 9.
When the equation $3 a + 6 = 15$ has 6 subtracted from both sides and becomes $3 a = 9$ , then both sides are divided by 9, the equivalent equation is $\frac{1}{3} a = 1$ .

Are you ready for more?

Extending Your Thinking

Here is a puzzle:

$m + m = N$
$N + N = p$
$m + p = Q$
$p + Q = ?$

Write two expressions that are equivalent to $p + Q$ .

Compare your answer: Your answer may vary, but here is a sample.

$2 p + m$
$9 m$

Self Check

Which equation is equivalent to

3x-6=18

$3x + 12 = 0$
$x - 2 =18$
$3x = 12$
$3x = 24$

Additional Resources

Finding Equivalent Equations

Equivalent equations are equations that have the same solutions. They are often found by using inverse operations or by multiplying each term in the equation by the same value.

Example

Write three equivalent equations to $4 x - 2 = 8$ using one of three strategies.

Using an inverse operation
Using division
Using multiplication

Using an inverse operation:
An equivalent equation can be determined by adding 2 to both sides.

$4 x - 2 = 8$
$4 x - 2 + 2 = 8 + 2$
$4 x = 10$
$(4 x - 2 = 8)$ and $(4 x = 10)$ are equivalent.

Using division:
An equivalent equation can be determined by dividing by 2 (or multiplying by $\frac{1}{2}$ ).

$4 x - 2 = 8$
$(\frac{1}{2}) (4 x - 2) = (\frac{1}{2}) (8)$
$2 x - 1 = 4$

$(4 x - 2 = 8)$ and $(2 x - 1 = 4)$ are equivalent.

Using multiplication:
An equivalent equation can be determined by multiplying the same number to both sides.

What is the equivalent equation if you multiply each term in the equation $4 x - 2 = 8$ by 3?

Compare your answer: $12 x - 6 = 24$ , because:

$4 x - 2 = 8$
$(3) (4 x - 2) = (3) (8)$
$12 x - 6 = 24$

Try it

Finding Equivalent Equations

Determine the equivalent equation if you subtract 5 from each side of the equation, $3 x + 5 = 15$ .

Compare your answer:

$3 x = 10$

Explain how $(3 x = 10)$ is related to $(x = \frac{10}{3})$ .

Compare your answer:

Divide both sides by 3. $\frac{3 x}{3} = \frac{10}{3}$ so $x = \frac{10}{3}$

Multiply every term by the same number to determine an equivalent equation to $3 x + 5 = 15$ .

Compare your answer:

Your answer may vary, but here are some samples.

$2 (3 x + 5 = 15)$ becomes $6 x + 10 = 30$
$4 (3 x + 5 = 15)$ becomes $12 x + 20 - 60$
$- 2 (3 x + 5 = 15)$ becomes $- 6 x - 10 = 30$

Is $(3 x + 5 = 15)$ equivalent to $(6 x + 15 = 60)$ ?

No.

Explain your reasoning for your answer to number 4 and whether the equations $(3 x + 5 = 15)$ and $(6 x + 15 = 60)$ are equivalent.

Compare your answer: Your answer may vary, but here is a sample:

No. Each term in the first equation was multiplied by a different number to get equation two.