일차방정식

$104^\circ \,\text{F}$

240 ft

Answers will vary.

84 adult tickets, 31 student tickets

615 children’s tickets and 195 adult tickets

Orlando mixed five pounds of cereal squares and 25 pounds of nuts.

Becca mixed 21 gallons of fruit punch and seven gallons of soda.

Step 1. Read the problem. Make sure all the words and ideas are understood.

• Draw a diagram to illustrate what it happening. Shown below is a sketch of what is happening in the example.

• Create a table to organize the information.
• Label the columns “Rate,” “Time,” and “Distance.”
• List the two scenarios.
• Write in the information you know.

Step 2. Identify what you are looking for.

You are asked to find the speed of both bikers.

Notice that the distance formula uses the word “rate,” but it is more common to use “speed”

when we talk about vehicles in everyday English.

Step 3. Name what we are looking for. Choose a variable to represent that quantity.

• Complete the chart
• Use variable expressions to represent that quantity in each row. We are looking for the speed of the bikers. Let’s let r represent Wayne’s speed. Since Dennis’ speed is 7 mph faster, we represent that as $r+7$ $\begin{array}{ccc}r+7 & = & \text{Dennis’ speed} \\ r & = & \text{Wayne’s speed}\end{array}$ Fill in the speeds into the chart.
• Multiply the rate times the time to get the distance.

Step 4. Translate into an equation.

• Restate the problem in one sentence with all the important information.
• Then, translate the sentence into an equation. The equation to model this situation will come from the relation between the distances. Look at the diagram we drew above. How is the distance travelled by Dennis related to the distance travelled by Wayne? Since both bikers leave from Riverside and travel to the beach, they travel the same distance. So we write:

Step 5. Solve the equation using algebra techniques.

Now solve this equation.	So Wayne’s speed is 21 mph.
Find Dennis’ speed.	Dennis’ speed 28 mph.

Step 6. Check the answer in the problem and make sure it makes sense.

$\begin{array}{cc}\text{Dennis} & 28\,\text{mph}(1.5\,\text{hours})=42\,\text{miles} \\ \text{Wayne} & 21\,\text{mph}(2\,\text{hours})=42\,\text{miles}\checkmark \end{array}$

Step 7. Answer the question with a complete sentence.

Wayne rode at 21 mph and Dennis rode at 28 mph.

The speed of the local train is 48 mph and the speed of the express train is 60 mph.

Jeromy drove at a speed of 80 mph and his mother drove 60 mph.

Step 1. Read the problem. Make sure all the words and ideas are understood.

• Draw a diagram to illustrate what it happening. Below shows a sketch of what is happening in the example.
• Create a table to organize the information.
• Label the columns rate, time, distance.
• List the two scenarios.
• Write in the information you know.

Step 2. Identify what we are looking for.

We are asked to find the average speeds of Carina and her brother.

Step 3. Name what we are looking for. Choose a variable to represent that quantity.

• Complete the chart.
• Use variable expressions to represent that quantity in each row. We are looking for their average speeds. Let’s let r represent the average speed of Carina's brother. Since Carina’s speed is 15 mph faster, we represent that as $r+15.$ Fill in the speeds into the chart.
• Multiply the rate times the time to get the distance.

Step 4. Translate into an equation.

• Restate the problem in one sentence with all the important information.
• Then, translate the sentence into an equation. Again, we need to identify a relationship between the distances in order to write an equation. Look at the diagram we created above and notice the relationship between the distance Carina traveled and the distance her brother traveled. The distance Carina traveled plus the distance her brother travel must add up to 410 miles. So we write:

Step 5. Solve the equation using algebra techniques.

Now solve this equation.	So Carina’s brother's speed was 50 mph.
Carina’s speed is $r+15.$	Carina’s speed was 65 mph.

Step 6. Check the answer in the problem and make sure it makes sense. $\begin{array}{cccc}\text{Carina drove} & 65\,\text{mph}(3\,\text{hours}) & = & \quad 195\,\text{miles} \\ \text{Her brother drove} & 50\,\text{mph}(4\,\text{hours}) & = & \underset{-----}{200\,\text{miles}} \\ & & & \quad 395\,\text{miles}\checkmark \end{array}$

Step 7. Answer the question with a complete sentence.

Carina drove 65 mph and her brother 50 mph.

Christopher’s speed was 50 mph and his parents’ speed was 40 mph.

Ashley’s parents drove 55 mph and Ashley drove 62 mph.

Step 1. Read the problem. Make all the words and ideas are understood.

• Draw a diagram to illustrate what it happening.
• Create a table to organize the information.
• Label the columns rate, time, distance.
• List the two scenarios.
• Write in the information you know.

Step 2. Identify what we are looking for.

We are asked to find the amount of time the trucks will travel until they are 325 miles apart.

Step 3. Name what we are looking for. Choose a variable to represent that quantity.

• Complete the chart.
• Use variable expressions to represent that quantity in each row. We are looking for the time travelled. Both trucks will travel the same amount of time. Let’s call the time t. Since their speeds are different, they will travel different distances.
• Multiply the rate times the time to get the distance.

Step 4. Translate into an equation.

• Restate the problem in one sentence with all the important information.
• Then, translate the sentence into an equation. We need to find a relation between the distances in order to write an equation. Looking at the diagram, what is the relationship between the distances each of the trucks will travel? The distance travelled by the truck going west plus the distance travelled by the truck going east must add up to 325 miles. So we write:

Step 5. Solve the equation using algebra techniques.

$\quad \begin{array}{cccccc}\text{Now solve this equation} & & & 70t+60t & = & 325 \\ & & & 130t & = & 325 \\ & & & t & = & 2.5\end{array}$

So it will take the trucks $2.5$ hours to be 325 miles apart.

Step 6. Check the answer in the problem and make sure it makes sense.

$\quad \begin{array}{cccc}\text{Truck going West} & 70\,\text{mph}(2.5\,\text{hours}) & = & \,175\,\text{miles} \\ \text{Truck going East} & 60\,\text{mph}(2.5\text{hours}) & = & \underset{-----}{150\,\text{miles}} \\ & & & \,325\,\text{miles}\checkmark \end{array}$

Step 7. Answer the question with a complete sentence. It will take the trucks 2.5 hours to be 325 miles apart.

Pierre and Monique will be 429 miles apart in 3 hours.