## Math 9 Chapter 4 Lesson 8: Solve math problems by making equations

## 1. Summary of theory

### 1.1. Solution method

To solve a problem by making an equation, we follow these steps:

**Step 1:** Equation

- Select hide and set conditions for hiding
- Expressing different quantities implicitly
- Based on the problem, make an equation in the form you have learned

**Step 2:** Solve the equation

**Step 3:** Compare the results and choose the appropriate solution

### 1.2. Basic math forms

- Motion math form
- Math form combining geometric quantities
- Math form working together as a team, working individually
- Math form of flowing water
- Math form to find numbers
- Math form combines physics, chemistry

## 2. Illustrated exercise

### 2.1. Basic exercises

**Question 1:** A rectangular piece of land has a width of 4 m less than its length and an area of 320 m .^{2}. Calculate the length and width of the plot.

**Solution guide**

Let the length of the garden be x (m) (x > 4)

The width of the garden is x – 4 (m)

The area of the garden is 320m .^{2} so we have the equation:

\(\eqalign{& x\left( {x – 4} \right) = 320 \cr & \Leftrightarrow {x^2} – 4x – 320 = 0 \cr & \Delta ‘ = {2^2} + 320 = 324;\,\,\sqrt {\Delta ‘} = 18 \cr & {x_1} = 2 + 18 = 20;\,\,{x_2} = 2 – 18 = – 16 \cr} \)

\(x_2 = -16\) does not satisfy the implicit condition.

So the length of the garden is 20m

The width of the garden is 16 m .

**Verse 2:** You go from A to B 120 km apart in the planned time. After 1 hour, Thu rests for 10 minutes, so in order for Thu to arrive at B on time, she has to increase her speed by \(6km/h\). Calculate Thu’s initial velocity.

**Solution guide**

Let Thu’s initial speed be \(x(km/h); (x>0)\)

The intended time to B is \(\frac{120}{x}\)(h)

After 1 hour, Thu has traveled x km, the remaining distance is \(120-x\)

Time to travel the remaining distance is \(\frac{120-x}{x+6}\)(h)

We have the equation: \(\frac{120}{x}=1+\frac{10}{60}+\frac{120-x}{x+6}\)

\(\Leftrightarrow x^2+42x-4320=0\)

\(x=48\) (receive)

\(x=-90\) (type)

### 2.2. Advanced exercises

**Question 1:** During group study, friend Hung asked Minh and Lan to each choose a number so that the difference between these two numbers is 5 and their product must be equal to 150. What numbers should Minh and Lan have to choose?

**Solution guide**

Call the number you chose: \(x\) and the number you chose: \(x+5\).

The product of two numbers is: \(x(x+5)\)

At the beginning of the article, we have the equation:

\(x(x+5)=150\) or \({x^2}+5x-150=0\)

Solving the equation we get: \(\Delta = {5^2} – 4.1.( – 150) = 625 > 0\)

Then the equation has 2 solutions: \({x_1}= \dfrac{{ – 5 + \sqrt {625} }}{2}=10,\)\({x_2} = \dfrac{{ – 5 – \sqrt {625} }}{2}=-15\)

So:

If Minh chooses number 10, Lan chooses number 15 or vice versa.

If Minh chooses the number -15, then Lan chooses the number -10 or vice versa.

**Verse 2: **A team of workers completes a work of 420 products. If the team increases by 5 people, the number of working days will be reduced by 7 days. Find the number of workers.

**Solution guide**

Let’s call the number of workers \(x(x>0;x\epsilon \mathbb{N})\)

Number of days completed vs x people is \(\frac{420}{x}\) (days)

The number of workers after increment is: \(x+5\)

The new completion date is \(\frac{420}{x+5}\) (days)

We have the following equation: \(\frac{420}{x}-\frac{420}{x+5}=7\)

Solving the above equation we get

\(x=15\) (receive)

\(x=20\) (type)

So the number of workers is 15 people

## 3. Practice

### 3.1. Essay exercises

**Question 1: **Given a two-digit number. The sum of their two digits is \(10\). The product of those two digits is less than the given number \(12\). Find the given number.

**Verse 2: **In a meeting room there are \(360\) chairs arranged in rows and the number of seats in each row is equal. There was a time when the meeting room had to add a row of seats and each row increased \(1\) seats (the number of seats in the rows remained the same) to have enough room for \(400\) delegates. How many rows of chairs do you normally have in a room?

**Question 3: **A transport company intends to use a large vehicle to carry \(15 \) tons of vegetables under a contract. But when entering the job, the company no longer had large vehicles, so it had to be replaced with vehicles with a tonnage of less than half a ton. In order to ensure the contracted time, the company must use a larger number of vehicles than the planned number of \(1\) vehicles. How many tons is the tonnage of each small truck?

**Question 4: **A concrete batching plant must produce \(450m^3\) concrete for an irrigation dam in a specified time. Thanks to increased productivity every day \(4,5m^3\) so \(4\) days before the specified time limit, the team has produced \(96\% \) jobs. How many days is the allotted time?

**Question 5: **People mix \(8g\) this liquid with \(6g\) another liquid with a density less than \(0.2g/cm^3\) to get a mixture with a density of \( 0.7g/cm^3\). Find the density of each liquid.

### 3.2. Multiple choice exercises

**Question 1: **Find two numbers whose sum is 19 and sum of squares is 185.

A. \(13;6\)

B. \(12;7\)

C. \(11;8\)

D. \(10;9\)

**Verse 2: **The area of a right triangle whose hypotenuse is 13 cm and the sum of the two sides is 17 cm is:

A. \(15cm^2\)

B. \(30cm^2\)

C. \(45cm^2\)

D. \(60cm^2\)

**Question 3: **Find two numbers whose sum is 8 and their product is 15

A. \((-3;-5);(-5;-3)\)

B. \((-2;-6);(-6;-2)\)

C. \((3;5);(5;3)\)

D. \((2;6);(6;2)\)

**Question 4: **A fleet of trucks carrying 168 tons of paddy. If 6 trucks are increased and 12 tons of paddy are loaded, each truck will be 1 ton lighter than the original. How many cars were there in the beginning?

A. 24 cars

B. 25 cars

C. 26 vehicles

D. 27 cars

**Question 5: **1 Canoe 42 km downstream and then 20 km upstream for a total of 5 hours. The flow speed is 2 km/h. The speed of the canoe in still water is:

A. \(16km/h\)

B. \(14km/h\)

C. \(12km/h\)

D. \(10km/h\)

## 4. Conclusion

Through this lesson, students need to:

- State the steps to solve a problem by formulating an equation.
- Apply steps to solve problems.
- Reasoning, presenting scientific solutions, concisely, carefully and accurately.
- Flexible application of problem solving knowledge.

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