Set up and solve a practical linear equation

Combined total

\(\displaystyle 50\text{ parts}\)

Extra parts packed by Cedar

\(\displaystyle 8\text{ parts}\)

Two robotics teams pack spare parts. Cedar packs more than Ash. Let x be the number Ash packs. Which equation represents their combined total?

Write each team's amount before adding them

A workshop pays an $18 fixed charge plus $7 per kit. Its budget is $100. Predict the largest affordable whole count

Fixed charge

\(\displaystyle \$18\)

Charge per kit

\(\displaystyle \$7\)

Budget

\(\displaystyle \$100\)

A workshop pays an $18 fixed charge plus $7 per kit, with a $100 budget. Change the kit count. Find the last setting within the budget and the first setting over it

Which two settings straddle the budget?

Build the equation from the quantities

Choose one unknown and state what it measures. Express the other quantities using it. Use the total or equality in the description to form an equation, then use the solving methods you already know

1. 1Express each related amount Use Ash's count for the unknown. Add the stated difference to express Cedar's larger count \(\displaystyle \begin{aligned}\text{Ash}&:x\\\text{Cedar}&:x+8\end{aligned}\)
2. 2Use the relationship Add both expressions to match the combined total. Keep the brackets until the quantities have been combined \(\displaystyle \begin{aligned}x+(x+8)&=50\\2x+8&=50\end{aligned}\)
3. 3Preserve a grouped calculation When a description adds an amount before multiplying, bracket the changed number. Multiplying the original number first models a different order \(\displaystyle 4(x+2)=40\)
4. 4Set equal results equal Write an expression for each stated calculation using the same unknown. If their results agree, set the expressions equal \(\displaystyle 3x+7=2x+15\)
5. 5Check the situation After solving, check the total and the stated difference. Give the answer for the quantity the question asks for, with its unit
6. 6Treat a budget as a limit When total cost increases with each item and the budget is an upper limit, the maximum affordable whole count is the greatest integer at or below the exact budget boundary. Check that this count fits and the next count costs too much. Other contexts can require a different decision

Width

\(\displaystyle w\text{ m}\)

Length

\(\displaystyle (2w+3)\text{ m}\)

Perimeter

\(\displaystyle 54\text{ m}\)

A rectangular shade panel has the dimensions described. Find its width in metres

Write the sum of all four sides before simplifying it

First multiplier

\(\displaystyle 5\)

Amount subtracted

\(\displaystyle 9\)

Second multiplier

\(\displaystyle 2\)

Amount added

\(\displaystyle 12\)

Apply both rules to the same starting number. The first rule multiplies the number and then subtracts an amount. The second multiplies the number and then adds an amount. Their results are equal. Find the starting number

Express each result using the same unknown, then set the results equal

Delivery charge

\(\displaystyle \$22\)

Price per roll

\(\displaystyle \$9\)

Budget

\(\displaystyle \$108\)

A repair club buys complete tool rolls. The order has one delivery charge. What is the greatest number of rolls it can buy within its budget?

First work out how much of the budget remains for the rolls

Total trays

\(\displaystyle 62\)

How many fewer on the right

\(\displaystyle 6\)

Two benches hold trays. The right bench has fewer trays than the left. A student lets x be the number on the left. Where is the first error in the attempt?

Start with the left bench's count and remove the difference to describe the right bench

Team A’s share

\(\displaystyle x\)

Team B’s share

\(\displaystyle 2x-5\)

Team C’s share

\(\displaystyle x+9\)

Three teams share 84 parts. Which equation uses the total?

Equation or model

\(\displaystyle x+(2x-5)+(x+9)=84\)

Team A receives x parts. Find Team A’s share from the given equation

Equation or model

\(\displaystyle B=2(20)-5\)

Team A has 20 parts. Team B has five fewer than twice A’s share. Find B’s share

Equation or model

\(\displaystyle C=20+9\)

Team A has 20 parts and team C has nine more. Find C’s share

Five identical packs must contain exactly 42 items altogether. Each pack must have a whole number of items. What follows?

A 7 m ribbon is cut into four equal lengths, with no material lost. Which length is appropriate?

Equation or model

\(\displaystyle C=25+7n\)

The workshop’s fixed charge rises from $18 to $25 while each kit still costs $7 and the budget remains $100. Find the new maximum whole count

A workshop charges $25 plus $7 per kit and has a $100 budget. Which report correctly describes the maximum affordable whole number of kits?