Question

1. a seventh grade class and 7 adults go on a field trip. two thirds of the people ride on a bus. everyone else rides in vans. if 34 people ride on the bus, how many seventh graders go on the field trip? write and solve an equation. check your solution.

solving equations
for problems 4–6, solve for x. check your solution

1. $-\frac{3}{4}x - 12 = 34$

Explanation:

Problem 3 (Field Trip)

Step 1: Define the variable

Let \( x \) be the number of seventh - graders. The total number of people on the field trip is \( x + 7 \) (since there are \( x \) seventh - graders and 7 adults).

Step 2: Set up the equation

We know that two - thirds of the people ride on the bus, and the number of people on the bus is 34. So, the equation is \( \frac{2}{3}(x + 7)=34 \).

Step 3: Solve the equation

First, multiply both sides of the equation by \( \frac{3}{2} \) to get rid of the fraction on the left - hand side:
\( x + 7=34\times\frac{3}{2} \)
\( 34\times\frac{3}{2}=\frac{34\times3}{2}=51 \), so \( x + 7 = 51 \).
Then, subtract 7 from both sides of the equation:
\( x=51 - 7 \)
\( x = 44 \)

Step 4: Check the solution

The total number of people is \( 44+7 = 51 \). Two - thirds of 51 is \( \frac{2}{3}\times51 = 34 \), which matches the number of people on the bus. So the solution is correct.

Step 1: Add 12 to both sides of the equation

To isolate the term with \( x \), we add 12 to both sides.
\( \frac{3}{4}x-12 + 12=34 + 12 \)
\( \frac{3}{4}x=46 \)

Step 2: Solve for \( x \)

Multiply both sides of the equation by \( \frac{4}{3} \) to solve for \( x \).
\( x = 46\times\frac{4}{3}=\frac{184}{3}\approx61.33 \) (If we want an exact form, \( x=\frac{184}{3} \); if we want a decimal form, \( x\approx61.33 \))

Step 3: Check the solution

Substitute \( x = \frac{184}{3} \) into the left - hand side of the original equation:
\( \frac{3}{4}\times\frac{184}{3}-12 \)
First, \( \frac{3}{4}\times\frac{184}{3}=\frac{184}{4}=46 \)
Then, \( 46-12 = 34 \), which is equal to the right - hand side of the original equation. So the solution is correct.

Answer:

The number of seventh - graders on the field trip is 44.

Problem 4 (Solving the Equation \( \frac{3}{4}x-12 = 34 \))