Informative line

# Solving Models

• To solve a given model, first we should convert it into an algebraic equation.
• We can write algebraic equations through models.
• Consider an example to understand it.

Write an algebraic equation for the following model.

$$\triangle\triangle\triangle\bigcirc\bigcirc=\bigcirc\bigcirc\bigcirc\bigcirc\triangle$$...............(1)

• To write an algebraic equation, follow the given steps:

Step 1: Allot different variables to different figures.

Let $$\triangle=x\;\;\;\bigcirc=y$$

Step 2: Write individual expressions for both the groups that are present on both the sides of the equals sign.

For      $$\triangle\triangle\triangle\bigcirc\bigcirc$$

It is the group of three triangles 'and' two circles.

$$\therefore$$ Expression for this:

$$3x+2y$$     $$(\triangle=x,\;\;\;\bigcirc=y)$$

Similarly,         $$\bigcirc\bigcirc\bigcirc\bigcirc\triangle$$

It is the group four circles 'and' one triangle

$$\therefore$$ Expression for this:

$$4y+x$$        $$(\triangle=x,\;\;\;\bigcirc=y)$$

Step 3: Now put the expressions of both the groups in place of model to get an algebraic equation.

Thus, algebraic equation for the given model is:

$$3x+2y=4y+x$$

• Now solve the algebraic equation for $$x$$ and  $$y.$$

$$3x+2y=4y+x$$

Step 1: Since, $$x$$ being added to $$4y$$ therefore we will use inverse operation and subtract $$x$$ from both the sides.

$$3x+2y-x=4y+x-x$$

$$2x+2y=4y$$

Step 2: Since, $$2y$$ being added on both sides therefore we will use inverse operation and subtract $$2y$$ from both the sides.

$$2x+2y-2y=4y-2y$$

$$2x=2y$$

or $$\dfrac{2x}{2}=\dfrac{2y}{2}$$

$$x=y$$

Step 3: Replace the values of $$x$$ and $$y$$.

Thus, $$\triangle=\bigcirc$$

#### An equation is modeled as shown below: Which one of the following options is correct?

A

B

C

D

×

Given model equation:

To solve a given model, first we should convert it into an algebraic equation.

Allot different variables to different figures.

Let

$$\bigcirc=x$$ and $$\Box=y$$

Write individual expressions for both the groups that are present on both the sides of the equals sign.

For

It is a group of three circles 'and' three square.

$$\therefore$$ Expression for this:

$$3x+3y$$        $$(\bigcirc=x,\;\;\;\Box=y)$$

For

It is a group of two squares 'and' five circles.

$$\therefore$$ Expression for this:

$$2y+5x$$        $$(\Box=y,\;\;\;\bigcirc=x)$$

Now put the expressions of both the groups in the given model to get an algebraic equation.

Thus, algebraic equation for the given model is:

$$3x+3y=2y+5x$$

Solving this algebraic equation for $$x$$ and $$y.$$

Since, $$3x$$ is being added to $$3y$$ therefore we will use inverse operation and subtract $$3x$$ from both the sides.

$$3x+3y-3x=2y+5x-3x$$

$$3y=2y+2x$$

Since, $$2y$$ is being added to $$2x$$ therefore, we will use inverse operation and subtract $$2y$$ from both the sides.

$$3y-2y=2y+2x-2y$$

$$y=2x$$

Replace the values of $$x$$ and $$y.$$

Thus,  $$\Box=2\bigcirc$$

or

Hence, option (A) is correct.

### An equation is modeled as shown below: Which one of the following options is correct?

A
B
C
D

Option A is Correct

# Solving an Equation

• Solving an equation means to find the value of a variable.
• To solve an equation, we use the inverse operations to get only the variable on one side of the equation.
• An inverse operation means opposite of an operation.
• Inverse operation of addition is subtraction and vice versa.
• Inverse operation of multiplication is division and vice versa.

For example:

Solve:  $$x+7=12$$

Here,

• We need to find the value of the variable $$(x)$$.
• Here, a variable $$(x)$$ is added to $$7$$.
• We will use inverse operation of addition, i.e. subtraction.

Subtract $$7$$ from both the sides to get only $$x$$ on the left-hand side.

$$x+7-7=12-7$$

$$\Rightarrow\;x+0=5$$

$$\Rightarrow\;x=5$$

• So, answer is $$5$$ or the value of $$x$$ is $$5$$.

#### Solve: $$3a+5=14$$

A $$3$$

B $$14$$

C $$5$$

D $$1$$

×

Given: $$3a+5=14$$

Here, $$5$$ is added to $$3a$$, so we will use inverse operation of addition, i.e. subtraction.

Thus, we subtract $$5$$ from both the sides of the equation.

$$3a+5-5=14-5$$

$$3a+0=9$$

$$3a=9$$

To get only $$'a'$$ on one side of the equation, we use inverse operation again.

Here, $$3$$ is multiplied by $$'a'$$, it means the inverse operation of multiplication, i.e. division should be used.

Thus, we divide by $$3$$ on both the sides of the equation.

$$\dfrac{3a}{3}=\dfrac{9}{3}$$

$$a=3$$

Thus, answer is $$3$$ or the value of a is $$3.$$

Hence, option (A) is correct.

### Solve: $$3a+5=14$$

A

$$3$$

.

B

$$14$$

C

$$5$$

D

$$1$$

Option A is Correct

# Comparison of Both Sides of an Equation

• We can determine the values of unknown constants by comparing both sides of the equation.
• In the equation, coefficients of same variable on both the sides are equal.
• We can determine the values of unknown constants by following the given steps:

Step-1 Identify the constants and variables.

Step-2 Compare the constant terms on both the sides of the equation.

Step-3 Compare coefficients of same variable on both the sides.

Example: $$ax+12=5x+b$$

If $$a$$ and $$b$$ are integers, determine their values.

• First, identify the constants and variables.

Constants : $$a,\;12,\;5,\;b$$

Variable : $$x$$

• Compare the constant terms on both the sides of the equation.

$$ax+12=5x+b$$

$$12=b$$ ... (1)

• Compare the coefficients of $$x$$ on both the sides

$$ax+12=5x+b$$

$$a=5$$ ... (2)

• From (1) and (2), we get:

$$a=5$$ and $$b=12$$

#### $$(A+1)x+(B-5)y=9x+y+C$$ Find the values of constants $$A,\;B$$ and $$C.$$

A $$A=1,\;B=5,\;C=0$$

B $$A=9,\;B=1,\;C=0$$

C $$A=1,\;B=5,\;C=1$$

D $$A=8,\;B=6,\;C=0$$

×

Given : $$(A+1)x+(B-5)y=9x+y+C$$

First, identify the constants and variables.

Constants : $$(A+1),\;(B-5),\;9,\;C$$

Variables : $$x$$ and $$y$$

Compare the constant terms on both the sides.

$$(A+1)x+(B-5)y=9x+y+C$$

$$0=C$$  ... (1)

Compare the coefficients of $$x$$ on both the sides.

$$(A+1)x+(B-5)y=9x+y+C$$

$$A+1=9$$

$$\Rightarrow\;A=8$$  ... (2)

Compare the coefficients of $$y$$ on both the sides.

$$(A+1)x+(B-5)y=9x+y+C$$

$$B-5=1$$

$$\Rightarrow\;B=6$$  ... (3)

From (1), (2) and (3), we get :

$$A=8,\;B=6,\;C=0$$

Hence, option (D) is correct.

### $$(A+1)x+(B-5)y=9x+y+C$$ Find the values of constants $$A,\;B$$ and $$C.$$

A

$$A=1,\;B=5,\;C=0$$

.

B

$$A=9,\;B=1,\;C=0$$

C

$$A=1,\;B=5,\;C=1$$

D

$$A=8,\;B=6,\;C=0$$

Option D is Correct

# Evaluation of an Equation

• An equation in two variables can be evaluated if the value of one of the variables is known to us.

Steps to evaluate an equation:

• Put the value of the known variable in the given equation.
• Solve the equation for the other unknown variable.

For example:

In the equation, $$x+y=10$$;

what is the value of $$x$$  if  $$y=5$$ ?

• Put the value of the known variable, $$y=5$$,  in the equation,

$$x+y=10$$

$$\Rightarrow\;x+5=10$$

• Solve the equation to find $$x.$$

$$x+5=10$$

$$\Rightarrow x+5-5=10-5$$

$$\Rightarrow x=5$$

It is the required solution.

#### If the equation is $$2x+3y=15$$, what is the value of $$x$$  if  $$y=3$$?

A $$5$$

B $$7$$

C $$4$$

D $$3$$

×

To evaluate the equation,

$$2x+3y=15$$

Put the value of $$y=3$$ in the equation

$$2x+3y=15$$

$$\Rightarrow\;2x+3(3)=15$$

$$\Rightarrow\;2x+9=15$$

Solve the equation to find $$x.$$

$$2x+9=15$$

$$\Rightarrow\;2x+9-9=15-9$$

$$\Rightarrow\;2x=6$$

$$\Rightarrow\;\dfrac{2x}{2}=\dfrac{6}{2}$$

$$\Rightarrow\;x=3$$

This is the required solution.

Hence, option (D) is correct.

### If the equation is $$2x+3y=15$$, what is the value of $$x$$  if  $$y=3$$?

A

$$5$$

.

B

$$7$$

C

$$4$$

D

$$3$$

Option D is Correct

# Word Problems

• Word Problems can be solved by evaluating equations, where value of one variable is given and we have to find the value of other variable.
• We will explain the above statement with the help of an example.

For example: At present, age of father is $$18$$ more than twice the age of son. If father's age is $$38$$ years, calculate the age of his son at present. In the given situation, two variables are used:

Let $$x,$$ represents the age of son.

$$y,$$ represents the age of father.

Since, the age of the father is $$18$$ more than twice the age of the son.

$$\therefore\;\text{Age of father}=2(\text{age of son})+18$$

$$\Rightarrow\;y=2x+18$$

To find the age of the son when his father's age is $$38$$ years.

Put $$y=38$$ in the equation

$$38=2(x)+18$$

$$2x=38-18$$

$$2x=20$$

$$\Rightarrow \;x=10$$ years

This means, when the age of the father is $$38$$ years old, his son is $$10$$ years old.

Or,

Age of the father is $$18$$ more than twice the age of the son.

Let the age of son is $$x.$$

$$38-18=20$$

$$x+x=20$$

$$2x=20$$

$$x=10$$

Hence, son's age is $$10$$ years.

#### Tim went to a store to buy some T-shirt and trousers. The price of each T-shirt is $$4$$ less than the half the price of a trouser. Calculate the price of a T-shirt when the price of the trouser is $$50$$.

A $$21$$

B $$25$$

C $$22$$

D $$15$$

×

Given statement:

Price of a T-shirt is $$4$$ less than the half the price of a trouser.

Let $$a,$$ represents the price of each trouser.

$$b,$$ represents the price of each T-shirts.

According to the statement:

$$\text{Price of T-shirt}=\dfrac{\text{Price of trouser}}{2}-4$$

$$\Rightarrow\;b=\dfrac{a}{2}-4$$

To find the price of a T-shirt when price of the trouser is $$50$$,

We put $$a=50$$ in the equation:

$$\Rightarrow\;b=\dfrac{50}{2}-4$$

$$\Rightarrow\;b=25-4$$

$$\Rightarrow\;b=\,21$$

This means, the price of T-shirt is $$21$$.

Hence, option (A) is correct.

### Tim went to a store to buy some T-shirt and trousers. The price of each T-shirt is $$4$$ less than the half the price of a trouser. Calculate the price of a T-shirt when the price of the trouser is $$50$$.

A

$$21$$

.

B

$$25$$

C

$$22$$

D

$$15$$

Option A is Correct

# Equivalent Equation

• Two equations are equivalent if both give same results.
• When we solve an equation, every step of the solution is equivalent to each other.

For example: Solve the equation,

$$5x+8=18$$

$$5x+8=18$$ is equivalent to $$5x+8-8=18-8$$

$$5x+8=18$$ is equivalent to $$5x=10$$

$$5x+8=18$$ is equivalent to $$x=2$$

Examples:

1. $$x+5=20$$ in equivalent to $$x=15$$
2. $$3x-4=14$$ is equivalent to $$3x=18$$
3. $$2+(x+4)=9$$ is equivalent to $$(2+x)+4=9$$
• We can also find equivalent equations by using properties.
• For example, $$2x(4+1)=20$$

So, by using commutative property, the equivalent equation is: $$2x(1+4)=20$$

#### Which one of the following equations is equivalent to the equation:   $$2(2x-1)=14$$

A $$4x-4=14$$

B $$4x=14$$

C $$4x=16$$

D $$2x-2=14$$

×

Given: $$2(2x-1)=14$$

We can find the equivalent equation of the given equation by solving it.

$$2(2x-1)=14$$

$$4x-2=14$$

$$4x=16$$

We can see that option (C) is equivalent to the given equation.

Therefore $$2(2x-1)=14$$ is equivalent to $$4x=16$$

Hence, option (C) is correct.

### Which one of the following equations is equivalent to the equation:   $$2(2x-1)=14$$

A

$$4x-4=14$$

.

B

$$4x=14$$

C

$$4x=16$$

D

$$2x-2=14$$

Option C is Correct

# Substitution Method

• The substitution method is used to determine the value of a variable.
• For substitution method, the number of variables used in the equations should be equal to the number of equations.

For substitution method, we follow the given steps -

Step-1 Substitute the value of the given variable in the equation and determine the value of another variable.

Step-2 Again substitute the value of the variable (which was found out in step $$1$$) in next equation.

Step-3 Repeat the process to determine the value of each variable.

Example:

$$3x+2y=5,\;x+z=4,\;z=1$$

Determine the values of $$x$$ and $$y.$$

$$3x+2y=5$$  ... (1)

$$x+z=4$$  ... (2)

• Substitute $$z=1$$ in equation (2) and determine the value of $$x.$$

$$x+1=4$$

$$x=4-1$$

$$x=3$$  ... (3)

• Substitute $$x=3$$ in equation (1) and determine the value of $$y.$$

$$3(3)+2y=5$$

$$9+2y=5$$

$$2y=5-9$$

$$2y=-4$$

$$y=\dfrac{-4}{2}=-2$$

$$y=-2$$  ... (4)

• From equations (3) and (4), we get:

$$x=3$$

$$y=-2$$

#### $$4x+3y=16,\;x+z=3,\;z=2$$   Determine the values of $$x$$ and $$y.$$

A $$x=-1,\;y=4$$

B $$x=1,\;y=4$$

C $$x=1,\;y=-4$$

D $$x=-1,\;y=-4$$

×

Given:

$$4x+3y=16$$ ... (1)

$$x+z=3$$ ... (2)

$$z=2$$

Substitute $$z=2$$ in equation (2)

$$x+2=3$$

$$\Rightarrow\;x=1$$ ... (3)

Substitute  $$x=1$$ in equation (1)

$$4(1)+3y=16$$

$$\Rightarrow\;4+3y=16$$

$$\Rightarrow\;3y=16-4$$

$$\Rightarrow\;3y=12$$

$$\Rightarrow\;y=4$$ ... (4)

From equations (3) and (4), we get:

$$x=1$$

$$y=4$$

Hence, option (B) is correct.

### $$4x+3y=16,\;x+z=3,\;z=2$$   Determine the values of $$x$$ and $$y.$$

A

$$x=-1,\;y=4$$

.

B

$$x=1,\;y=4$$

C

$$x=1,\;y=-4$$

D

$$x=-1,\;y=-4$$

Option B is Correct

#### What is the value of $$y$$ in the given polynomial equation, if $$x = 3$$ ? $$y = x^3 + 6x+16$$

A $$60$$

B $$61$$

C $$72$$

D $$59$$

×

Given :

$$y =x^3 + 6x+16$$

$$x=3$$

We will put $$3$$ in place of $$x$$, to find the value of $$y$$.

$$y=x^3+6x+16$$

$$y=(3)^3+6(3)+16$$

$$y = 3 ×3×3+6×3+16$$

$$y = 27+18+16$$

$$y = 61$$

Hence, option (B) is correct.

### What is the value of $$y$$ in the given polynomial equation, if $$x = 3$$ ? $$y = x^3 + 6x+16$$

A

$$60$$

.

B

$$61$$

C

$$72$$

D

$$59$$

Option B is Correct