Forming & Solving Equations Flashcards

All 18 cards in this deck

What is the first step in forming an equation from a worded problem?

Define a letter for the unknown quantity.
e.g. "let $x$ be the number of books on shelf A"

How do you write "7 less than 4 times a number" as an expression?
e.g. call the number $n$

$4n-7$

How do you write three consecutive integers algebraically?
e.g. the smallest is $n$

$n$, $n+1$, $n+2$

What are the steps to form and solve a linear equation from a worded problem?
e.g. shelf A has $x$ books, shelf B has $(3x+1)$ books, and there are 45 books in total

1. Write the total as an expression and equate it to the given total: $x+3x+1=45$
2. Simplify and solve: $4x=44$, so $x=11$
3. Interpret: shelf A has 11 books

True or false? If a tea costs £$t$ and a coffee costs £$c$, then 3 teas and 2 coffees cost £$6tc$.

False.
The total cost is £$3t+2c$ — you multiply each price by its number and add.

How do you form a pair of simultaneous equations from a context?
e.g. 3 teas and 2 coffees cost £7.80; 5 teas and 4 coffees cost £14.20

Choose a letter for each unknown and write one equation for each piece of information: $3t+2c=7.80$ and $5t+4c=14.20$

What equation do you form when you are told the perimeter of a rectangle?
e.g. width $w$, length 7 cm longer, perimeter 70 cm

$2w+2(w+7)=70$
Add all four sides and set the total equal to the perimeter.

A pair of opposite sides of a rectangle is labelled with two different expressions. What equation can you form?
e.g. the sides are labelled $5x-9$ and $2x+6$

Equate the opposite sides: $5x-9=2x+6$
Opposite sides of a rectangle are equal in length.

What are the steps to form and solve an equation from the perimeter of a shape?
e.g. a triangle with sides $x$, $x+2$ and $x+4$ cm has perimeter 18 cm

1. Add the sides: $x+(x+2)+(x+4)=3x+6$
2. Set equal to the perimeter: $3x+6=18$
3. Solve and give units: $x=4$ cm

True or false? The area of a rectangle with sides $2x$ and $(x+3)$ is $2x+2(x+3)$.

False.
That is the perimeter; the area is the product $2x(x+3)$.

What equation do you form when told two shapes have areas related to each other?
e.g. square A has side $x$, square B has side $(x+4)$, and B's area is 70 cm² greater

Write an expression for each area and link them as stated: $(x+4)^2=x^2+70$

What are the steps when an area condition gives a quadratic equation?
e.g. a rectangle measures $x$ by $(x+3)$ cm and has area 10 cm²

1. Form the equation: $x(x+3)=10$
2. Rearrange to $ax^2+bx+c=0$: $x^2+3x-10=0$
3. Factorise and choose the valid root: $(x+5)(x-2)=0$, so $x=2$ cm ($-5$ rejected)

When an equation formed from a context is quadratic, how do you decide which root to use?

Reject any root that is impossible in the context, such as a negative length or a negative number of objects.

True or false? Once you have solved your equation and found $x$, the problem is finished.

False.
You must substitute $x$ back to find the quantity the question actually asks for.

What are the steps to solve a context problem with two unknowns?
e.g. 2 pens and 1 ruler cost £5; 1 pen and 1 ruler cost £3

1. Define letters and write two equations: $2p+r=5$, $p+r=3$
2. Eliminate one variable and solve: subtracting gives $p=2$
3. Substitute back and interpret: $r=1$, so a pen costs £2 and a ruler £1

What must you include with the final answer to a context problem?

The units and a clear statement of the quantity asked for.
e.g. "cost of one tea = £1.40"

How can you check a solution to an equation you formed from a worded problem?

Substitute the value back into the original situation and check it gives the stated total.

True or false? If Sam is $x$ years old and Tom is 3 years younger than Sam, then Tom's age is $(x-3)$ years.

True. "3 years younger" means subtract 3 from Sam's age, so Tom is $(x-3)$ years old.