Free preview 2/13

Question 1
A school tuck shop sells sandwiches and juice cartons. Two sandwiches and one juice cost $7, and one sandwich and one juice cost $4. If $x$ represents the price of a sandwich and $y$ the price of a juice, which pair of simultaneous equations models this situation?
No clue? Show me the answer
Correct answerCorrect!Incorrect
A $2 x + y = 7$ and $x + y = 4$
Step-by-step walkthrough
Choose a solution method
Method #1Direct Approach
Step 1: Assign variables
Let $x$ = price of a sandwich and $y$ = price of a juice carton, as stated in the question.
Step 2: Translate the first statement
"Two sandwiches and one juice cost $7" m e an s$ 2x + y = 7$.
Step 3: Translate the second statement
"One sandwich and one juice cost $4" m e an s$ x + y = 4$.
Step 4: Identify the correct pair
The correct pair of equations is $2 x + y = 7$ and $x + y = 4$ , which matches the first option.
Method #2Process of Elimination
Step 1: Understand what is being asked
We need the correct translation of both word statements into algebraic equations with $x$ = sandwich and $y$ = juice.
Step 2: Eliminate option with swapped coefficients
" $x + 2 y = 7$ and $x + y = 4$ " puts 2 with juice, not sandwiches. The first purchase has two sandwiches, so the coefficient 2 must be on $x$ . Eliminate this option.
Step 3: Eliminate option with swapped totals
" $2 x + y = 4$ and $x + y = 7$ " assigns $4 t o t h e p u r c ha se w i t h tw os an d w i c h es an d$ 7 to the one with one sandwich, which reverses the given totals. Eliminate this option.
Step 4: Eliminate option with doubled juice coefficient
" $2 x + 2 y = 7$ and $x + y = 4$ " suggests two juices were bought in the first purchase, but the problem states only one juice. Eliminate this option.
Step 5: Select the correct answer
The remaining option, " $2 x + y = 7$ and $x + y = 4$ ", correctly represents two sandwiches and one juice costing $7, an d o n es an d w i c han d o n e j u i cecos t in g$ 4.
Question 2
Given the simultaneous equations $y = 3 x - 1$ and $2 x + y = 9$ , what is the first substitution step when using the substitution method?
No clue? Show me the answer
Correct answerCorrect!Incorrect
AReplace $y$ with $3 x - 1$ in $2 x + y = 9$ to get $2 x + (3 x - 1) = 9$
Step-by-step walkthrough
Choose a solution method
Method #1Direct Approach
Step 1: Recognise the ready-made subject
The first equation already has $y$ isolated: $y = 3 x - 1$ . This is the ideal starting point for substitution.
Step 2: Substitute into the other equation
Replace every occurrence of $y$ in the second equation $2 x + y = 9$ with $3 x - 1$ , giving $2 x + (3 x - 1) = 9$ .
Step 3: Confirm the correct first step
The correct first substitution step is $2 x + (3 x - 1) = 9$ , which now contains only one variable and can be solved directly.
Method #2Process of Elimination
Step 1: Understand what is being asked
We need to identify the correct first step when applying the substitution method to these equations.
Step 2: Eliminate the incorrect rearrangement
"Replace $x$ with $y + 1$ " is not correct because $y = 3 x - 1$ does not rearrange to $x = y + 1$ ; the correct rearrangement would be $x = \frac{y + 1}{3}$ , which creates fractions unnecessarily.
Step 3: Eliminate multiplying both equations
"Multiply both equations by 3" is an elimination strategy step, not substitution, and is not needed here since $y$ is already isolated.
Step 4: Eliminate subtracting equations
"Subtract the two equations" is an elimination method step, not substitution. We should be replacing a variable, not combining equations.
Step 5: Select the correct answer
The correct first step is to replace $y$ with $3 x - 1$ in the second equation, giving $2 x + (3 x - 1) = 9$ .

Free preview

11 more questions in this topic

Pick this up in your Library: it holds the whole topic, notes, cheatsheet and questions. Your answers here are kept on this device.

Step-by-step walkthrough

Step 1: Assign variables

Step 2: Translate the first statement

Step 3: Translate the second statement

Step 4: Identify the correct pair

Step 1: Understand what is being asked

Step 2: Eliminate option with swapped coefficients

Step 3: Eliminate option with swapped totals

Step 4: Eliminate option with doubled juice coefficient

Step 5: Select the correct answer

Step-by-step walkthrough

Step 1: Recognise the ready-made subject

Step 2: Substitute into the other equation

Step 3: Confirm the correct first step

Step 1: Understand what is being asked

Step 2: Eliminate the incorrect rearrangement

Step 3: Eliminate multiplying both equations

Step 4: Eliminate subtracting equations

Step 5: Select the correct answer

11 more questions in this topic