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Linear Equations In Two Variables

Illustration Questions

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

Linear Equation

  • An equation in which the variables are raised to the first power and none of the terms is a product of two or more variables, is a linear equation.
  • When a linear equation contains two variables, it is called linear equation in two variables.

For example:

(i) \(x+y=12\)

(ii) \(y=10+x\)

In both the cases, the highest power of each variable is one.

Illustration Questions

Which one of the following represents a linear equation?

A \(x+y\)

B \(x^2+3=y^2\)

C \(x^2=9\)

D \(3x+4=y\)

×

When the variables in an equation are raised to the first power, the equation is a linear equation.

Option (A) is an expression, and not an equation because it does not have an equals sign.

Hence, option (A) is incorrect.

In options (B) and (C), the highest power of variables is \(2.\)

Hence, options (B) and (C) are incorrect.

In option (D), the variables are raised to the first power in the equation.

Hence, option (D) is correct.

Which one of the following represents a linear equation?

A

\(x+y\)

.

B

\(x^2+3=y^2\)

C

\(x^2=9\)

D

\(3x+4=y\)

Option D is Correct

Dependent and Independent Variables

Definition: 

  • An independent variable is a variable with a subject of choice, whose values can be changed. Its value is not affected by the values of another variable.
  • The value of a dependent variable depends upon the value of an independent variable. 
  • To identify a variable to be dependent or independent, fetch the important information from the given problem.

For example:

Alex earns \($10\) per hour for trimming lawns.

  • The amount of money he earns, \(m\) , depends upon how many hours he works for, \(h\).
  • The more hours he works for, the more money he earns. Therefore, the dependent variable is money, \(m\), and the independent variable is hours, \(h\).
  • The number of hours he works for does not depend upon the money he earns.

So, the independent variable \(=h\)

and the dependent variable \(=m\)

Illustration Questions

Tina is plucking flowers to make a bouquet. The more flowers she plucks, the bigger bouquet she will be able to make. Which one of following options represents the dependent variable?

A Number of flowers

B Size of bouquet

C Both (A) and (B)

D None of these

×

The size of the bouquet depends upon the number of flowers.

The more flowers she plucks, the bigger bouquet she will be able to make.

Thus, the dependent variable is the size of the bouquet.

Hence, option (B) is correct.

Tina is plucking flowers to make a bouquet. The more flowers she plucks, the bigger bouquet she will be able to make. Which one of following options represents the dependent variable?

A

Number of flowers

.

B

Size of bouquet

C

Both (A) and (B)

D

None of these

Option B 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.

Illustration Questions

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

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