Explore: Word Problems with Three Variables Solutions

When working with word problems, as always, first define the variables.
Then write a system of equations for a word problem.
Finally, solve the problem.

Example 4

Define your variables. Then write and solve the system of equations with three variables.

A catering company purchased shipments of fruits, grains, and vegetables for their next three events. Event A requires one crate of each and the total shipment weighed 41 pounds. Event B requires two crates of grains and two crates of vegetables for a total weight of 56 pounds. Event C requires two crates of vegetables, one crate of grains, and two crates of fruit, weighing 71 pounds. What are the individual weights of each crate?

Plan
Define the variables
Write a system of equations
Solve
Check

f : fruit

$A : f + g + v = 41$

g : grain

$B : 2 g + 2 v = 56$

v : veggies

$C : 2 f + g + 2 v = 71$

Note

You do not need to name the equation with a capital letter, but it may be helpful when determining the Plan.

Implement

$(2 g + 2 v = 56) (\frac{1}{2}) \Rightarrow g + v = 28$

$\begin{array}{rcl} f + g + v & = & 41 \\ f + (28) & = & 41 \\ f & = & 13 \end{array}$

$\begin{array}{rcl} 2 f + g + 2 v & = & 71 \\ 2 (13) + g + v + v & = & 71 \end{array}$

$\begin{array}{rcl} 2 (13) + (28) + v & = & 71 \\ 26 + 28 + v & = & 71 \\ v & = & 17 \end{array}$

$\begin{array}{rcl} (13) + g + (17) & = & 41 \\ g & = & 11 \end{array}$

$Check 13 + 11 + 17 = 41 ✓ 2 (11) + 2 (17) = 56 ✓ 2 (13) + 11 + 2 (17) = 71 ✓$

Explain

Multiply equation B by $\frac{1}{2} .$
Solve for f in equation A.
Substitute 28 for (g + v).
Substitute in the value of f and the value of (g + v) in equation C.
Solve for v.
Solve for g using equation A.
Substitute all values into each equation.
Check with a calculator.

The fruit crates weigh 13 pounds each, the grain crates weigh 11 pounds each, and the vegetable crates weigh 17 pounds each.

Note

Writing the solution in a sentence can help determine if the solution makes sense in context.

Example 5

Define your variables. Then write and solve the system of equations with three variables.

The average of three numbers is zero. The range for the set of numbers is eight. Three times the middle number, plus the smallest number, minus four times the largest number is –29. What are the three numbers in the data set?

x : smallest
$average : \frac{x + y + z}{3} = 0$

y : middle
$range : z - x = 8$

z : largest
$3 y + x - 4 z = - 29$

Implement

$\begin{array}{rcl} \frac{x + y + z}{3} & = & 0 \\ x + y + z & = & 0 \\ z - x & = & 8 \\ z & = & x + 8 \end{array}$

$\begin{array}{rcl} x + y + (x + 8) & = & 0 \\ 2 x + y & = & - 8 \end{array}$

$\begin{array}{rcl} x + 3 y - 4 (x + 8) & = & - 29 \\ x + 3 y - 4 x - 32 & = & - 29 \\ - 3 x + 3 y & = & 3 \end{array}$

$\begin{matrix} (2 x + y = - 8) (- 3) & \Rightarrow \end{matrix} \begin{array}{l} - 6 x - 3 y = 24 \\ + - 3 x + 3 y = 3 \\ - 9 x = 27 \\ x = - 3 \end{array}$

$\begin{array}{rcl} z & = & x + 8 \\ z & = & (- 3) + 8 \\ z & = & 5 \end{array}$

$\begin{array}{rcl} x + y + z & = & 0 \\ (- 3) + y + (5) & = & 0 \\ y + 2 & = & 0 \\ y & = & - 2 \end{array}$

The three numbers are –3, –2, and 5.

Explain

Multiplication Property of Equality
Solve for z in the range equation.
Substitute into average equation.
Substitute into third equation.
Combine equations from the previous two steps.
Solve for x
Solve for z
Solve for y

Note

Remember to check solutions by substituting the values back into every equation. If they are all true, the solution is correct. Using a calculator to check solutions may help you check answers more quickly.