Equations

UK Secondary (7-11)

Values that satisfy equations I

Lesson

"*Values that satisfy an equation*" is another way of saying "*numbers that make a number sentence true*." For example, if I wanted to find the value that satisfies the equation $x+1=3$`x`+1=3, we want to find a value for $x$`x` that makes that equation true. This statement is true when $x=2$`x`=2 because $2+1=3$2+1=3.

When we're working out whether or not a value satisfies the equation, we need to see whether the left hand side or the equation is the same as the right hand side. We can think of it as a see-saw.

For example, if I had the equation $x+12=20$`x`+12=20, we could think of it visually as:

If $12$12 was removed from the left hand side of the seesaw, it would look unbalanced like this:

So how do we balance the equation again? We need to remove $12$12 from the right hand side as well:

So $x=8$`x`=8 satisfies the equation $x+12=20$`x`+12=20 because $8+12=20$8+12=20.

Remember!

Whatever you do to one side of the equation, you must do to the other. This keeps your equation balanced.

Determine if $b=47$`b`=47 is a solution of $b+48=96$`b`+48=96.

a) Find the value of the left-hand side of the equation when $b=47$`b`=47.

Think: We need to substitute $47$47 into the equation for $b$`b`.

Do:

$\text{LHS }$LHS | $=$= | $b+48$b+48 |

$=$= | $47+48$47+48 | |

$=$= | $95$95 |

b) Is $b=47$`b`=47 a solution of $b+48=96$`b`+48=96?

Think: Is the LHS equal to the RHS in this equation?

Do: No, $b=47$`b`=47 is not a solution of $b+48=96$`b`+48=96.

John is paid $\$600$$600 a week as a carpenter. His friend Sean, who is paid $\$1200$$1200 a fortnight as a fireman, claims that he earns the same amount.

a) If weekly earnings is represented by $w$`w` and fortnightly earnings is represented by $f$`f`, which of the following equations represents the relationship between $w$`w` and $f$`f`?

A) $w=f-2$`w`=`f`−2 B) $f=\frac{w}{2}$`f`=`w`2 C) $2f=w$2`f`=`w` D) $2w=f$2`w`=`f`

Think: How many weeks are in a fortnight? Which equation represents that?

Do: There are two weeks in a fortnight. This is represented by equation D) $2w=f$2`w`=`f`.

b) Find John's fortnightly earnings. That is, using the equation $2w=f$2`w`=`f`, find the value of the left-hand side of the equation when $w=600$`w`=600.

Think: We need to substitute this value into the equation.

Do:

$2w$2w |
$=$= | $f$f |

$f$f |
$=$= | $2w$2w |

$=$= | $2\times600$2×600 | |

$=$= | $\$1200$$1200 |

Again, just like when we're working with equations, values that satisfy inequalities are values that make the inequality true. In you need a refresher about inequalities, click here.

Neville is saving up to buy a plasma TV that is selling for $\$950$$950. He has $\$650$$650 in his bank account and expects a nice sum of money for his birthday next month.

a) If the amount he is to receive for his birthday is represented by $x$`x`, which of the following inequalities models the situation where he is able to afford the plasma TV?

A) $x+650\le950$`x`+650≤950 B) $x+650\ge950$`x`+650≥950 C) $x-650\ge950$`x`−650≥950 D) $x-650\le950$`x`−650≤950

Think: How would we write an equation for the total amount of money Neville will have after his birthday? How much money will he need in his account to buy the TV?

Do:

After his birthday, Neville will have his $\$650$$650 plus an additional $x$`x` dollars from his parents.

We could write this algebraically as $650+x$650+`x` or $x+650$`x`+650.

He needs *at least* $\$950$$950 in his account to buy the TV.

Of the four options we have B) $x+650\ge950$`x`+650≥950 models the situation.

b) How much money would he have in total if his parents were to give him $\$310$$310 for his birthday?

Think: We found the algebraic expression for Neville's total amount of money in part a.

Do:

$\text{Total amount of money }$Total amount of money | $=$= | $310+650$310+650 |

$=$= | $\$960$$960 |

c) Would he have enough to buy the plasma TV if his parents were to give him $\$310$$310 for his birthday?

Think: What was the minimum amount of money Neville needs to buy the TV?

Do: He needs $\$950$$950 to buy the TV, so yes he would have enough money if his parents gave him $\$310$$310.

There are two rectangular-shaped pools at the local aquatic centre. Each pool has a length that is triple its width. Pool 1 has a perimeter of $256$256 metres.

Let the width of the pools be represented by $w$

`w`. Which of the following equations represents the perimeter of each pool in terms of $w$`w`?$2w+3w=256$2

`w`+3`w`=256A$3w+3w+3w=256$3

`w`+3`w`+3`w`=256B$w+3w=256$

`w`+3`w`=256C$w+w+3w+3w=256$

`w`+`w`+3`w`+3`w`=256D$2w+3w=256$2

`w`+3`w`=256A$3w+3w+3w=256$3

`w`+3`w`+3`w`=256B$w+3w=256$

`w`+3`w`=256C$w+w+3w+3w=256$

`w`+`w`+3`w`+3`w`=256DThe width of Pool 2 is $32$32 metres. Find its perimeter.

Is the width of Pool 1 also $32$32 metres?

no

Ayes

Bno

Ayes

B

We want to determine if $b=8$`b`=8 is the solution of $8b=63$8`b`=63.

Find the value of the left-hand side of the equation when $b=8$

`b`=8.Is $b=8$

`b`=8 the solution of $8b=63$8`b`=63?yes

Ano

Byes

Ano

B