Question 115·Easy·Linear Equations in Two Variables
A vending machine sells bottled water for $2 each and bottled juice for $3 each. Jamie buys a total of 8 bottles of water and juice for $18.
Let be the number of bottles of water Jamie buys and be the number of bottles of juice.
How many bottles of juice does Jamie buy?
For word problems that lead to linear equations in two variables, first define your variables clearly, then write one equation for a total count (like total items) and another for a total value (like total cost). Choose the simpler equation to solve for one variable and substitute into the other, reducing the system to a single linear equation. On the SAT, if the numbers are small and answer choices are given, you can also use a quick plug-in: test each choice for the target variable, compute the other variable from the total, and check which pair satisfies the value equation.
Hints
Identify the quantities and totals
You are told how many total bottles Jamie buys and how much total money Jamie spends. How can you write equations that use and to represent these totals?
Write the two equations
One equation should use and to represent the total number of bottles (8). The other should use the prices ($2 for water and $3 for juice) to represent the total cost ($18).
Use substitution or elimination
From the easier equation, solve for one variable (either or ) and substitute into the other equation. This will give you a single equation with one variable that you can solve.
Check with the choices if stuck
You can also plug each answer choice in for , compute , and check which pair gives a total cost of $18.
Desmos Guide
Enter the system of equations
In Desmos, treat as and as . In the first line, type x + y = 8. In the second line, type 2x + 3y = 18 to graph both lines.
Find the intersection of the lines
Look for the point where the two lines intersect. Tap or click that point so Desmos shows its coordinates .
Interpret the coordinates
The -coordinate is (water bottles) and the -coordinate is (juice bottles). Use the -value shown at the intersection to decide how many bottles of juice Jamie buys and pick the matching answer choice.
Step-by-step Explanation
Translate the word problem into equations
Let be the number of water bottles and be the number of juice bottles.
- Total bottles: Jamie buys 8 bottles in all, so
- Total cost: Water is $2 per bottle and juice is $3 per bottle, and the total cost is $18, so
Now we have a system of two equations with two variables.
Solve one equation for one variable
Use the simpler equation to express in terms of :
Now we can substitute into the cost equation.
Substitute and simplify to get an equation in
Substitute into :
Distribute the 2:
Combine like terms:
Now you have a simple one-step equation in terms of .
Solve for and match the answer choice
Solve by subtracting 16 from both sides:
So Jamie buys 2 bottles of juice, which corresponds to answer choice B) 2.