## Math 9 Chapter 3 Lesson 5: Solve the problem by making a system of equations

## 1. Summary of theory

### 1.1. Solution method

To solve the problem by making a system of equations, we follow these steps:

**Step 1:** Set up a system of equations

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

**Step 2:** Solve the system of equations

**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: **Solve the system of equations (I) and answer the given problem.

\(\left( I \right)\,\,\left\{ \matrix{- x + 2y = 1 \hfill \cr x – y = 3 \hfill \cr} \right.\)

**Solution guide:**

\(\left( I \right)\,\,\left\{ \matrix{- x + 2y = 1 \hfill \cr x – y = 3 \hfill \cr} \right \Leftrightarrow \left\{ \ matrix{y = 4 \hfill \cr x – y = 3 \hfill \cr} \right \Leftrightarrow \left\{ \matrix{y = 4 \hfill \cr x = 7 \hfill \cr} \right.\ )

So the number to find is \(74\)

**Verse 2:** Write expressions containing implicits representing the distance traveled by each vehicle, up to the time when the two cars meet again. From there, deduce the equation representing the hypothetical distance from TP. Ho Chi Minh to City. Can Tho is 189 km long.

**Solution guide**

The distance traveled by the bus until they meet is: \(\dfrac{9}{5}y\) (km)

The distance traveled by the trucks until they meet is: \(\dfrac{{14}}{5}x\) (km)

According to the assumption that the distance from TP. Ho Chi Minh City to Ho Chi Minh City. Can Tho is 189 km long, so we have the equation

\(\dfrac{{14}}{5}x + \dfrac{9}{5}y = 189\)

**Question 3:** Find a two-digit number, given that the units digit is 2 more than the tens digit, the product of the two digits is 7 more than their sum.

**Solution guide**

Call that number \(\bar{ab},(a,b\epsilon \mathbb{N})\)

According to the problem, we have a system of equations: \(\left\{\begin{matrix} a+2=b\\ ab=a+b+7 \end{matrix}\right.\)\(\Rightarrow \left \{\begin{matrix} a=3\\ b=5 \end{matrix}\right.\)

So, the number to find is 35

### 2.2. Advanced exercises

**Question 1:** Find a three-digit number, knowing that when the number is divided by 11 the quotient is equal to the sum of the digits of the divisor.

**Solution guide:**

Call the search number \(\bar{abc}(a,b,c>0; a,b,c \epsilon \begin{Bmatrix} 1;10 \end{Bmatrix})\)

According to the problem, we have: \(100a+10b+c=11(a+b+c)\)

\(\Leftrightarrow 100a+10b+c=11a+11b+11c\)

\(\Leftrightarrow 89a=b+10c\)

If \(a>1\Rightarrow 89a\) has at least 3 digits, the right-hand side is a two-digit sum.

So \(a=1\)\(\Rightarrow 89=10c+b\)

Where \(10c+b\) is \(\bar{cb}\).

So the number to find is 198

**Verse 2: **Multiply a two-digit number by the sum of the digits and the result is 405. If written in the same way, the product is 468. Find that number.

__Solution guide:__

Call the search number \(\bar{ab}(a;b\epsilon \mathbb{N})\)

According to the problem, we have a system of equations \(\left\{\begin{matrix} (10a+b).(a+b)=405\\ (10b+a).(b+a)=486 \end{ matrix}\right.\)

\(\Leftrightarrow \left\{\begin{matrix} 10a^2+11ab+b^2=405(1)\\ 10b^2+11ab+a^2=486(2) \end{matrix}\right .\)

Subtract (2) from (1) to get: \(b^2-a^2=9\Leftrightarrow (ba)(a+b)=9\)

Where a, b are natural numbers, it is easy to deduce \(a=4;b=5\)

So the number to find is 45

## 3. Practice

### 3.1. Essay exercises

**Question 1:** Find two natural numbers, know that their sum is 1006 and if the large number is divided by the smaller number, the quotient is 2 and the remainder is 124.

**Verse 2:** A car starts from A and is expected to arrive at B at 12 noon. If the car is traveling at 35 km/h, it will arrive at B 2 hours later than prescribed. If the car is traveling at 50 km/h, it will arrive at B 1 hour earlier than prescribed. Find the length of the distance AB and the starting time of the car at A.

**Question 3: **The sum of two numbers is 59. Twice of one is three times less than the other is 7. Find those two numbers.

**Question 4: **Seven years ago the mother’s age was five times the son’s age plus 4.

This year, the mother’s age is exactly three times that of her son’s age.

How old is each person this year?

### 3.2. Multiple choice exercises

**Question 1:** A rectangular piece of land has an area of \(300(m^2)\), if the width is increased by \(5m\) and the length is decreased by \(5m\), the area remains the same. The perimeter of the original plot is:

A. \(60m\)

B. \(65m\)

C. \(70m\)

D. \(75m\)

**Verse 2: **There are two boxes of marbles, if you take the number of balls from the first box as many balls as the number of balls in the second box and put them in the second box, then take from the second box as many balls as the number of balls left in the first box and discard them. put in the first box, finally take from the first box the number of marbles equal to the number of balls left in the second box and put it in the second box, we get 16 balls in each box. The initial number of marbles in the boxes is:

A. \(24;8\)

B. \(22;10\)

C. \(20;12\)

D. \(18;14\)

**Question 3:** A rectangular garden has perimeter \(48m\). If the width is increased 4 times and the length is increased 3 times, the perimeter of the garden becomes \(162m\). The original garden area is:

A. \(72 (m^2)\)

B. \(144 (m^2)\)

C. \(216 (m^2)\)

D. \(288 (m^2)\)

**Question 4: **Find a two-digit number, knowing that the ones digit is 3 less than the tens digit, and the product of those two digits is 27 greater than their sum.

A. \(58\)

B. \(85\)

C. \(69\)

D. \(96\)

**Question 5: **Multiplying a two-digit number by the sum of the digits gives the product 684. If the number written by two numbers in reverse order is multiplied by the sum of the digits, the product is 900. to be:

A. \(86\)

B. \(68\)

C. \(75\)

D. \(57\)

## 4. Conclusion

This lesson helps students:

- Understand and understand the steps to solve problems by formulating a system of equations.
- Practice skills to solve systems of equations. Apply to solving specific problems

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