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LINEAR EQUATIONS IN TWO VARIABLES - Q&A

Exercise 4.1

1. The cost of a notebook is twice the cost of a pen. Write a linear equation in two variables to represent this statement. (Take the cost of a notebook to be ₹ x and that of a pen to be ₹ y).

Let the cost of a notebook be = ₹ x
Let the cost of a pen be = ₹ y
According to the problem, the cost of a notebook is twice the cost of a pen.
So, x = 2 × y
x = 2y
Subtracting 2y from both sides, we get:
x - 2y = 0
This is the required linear equation in two variables.

2. Express the following linear equations in the form ax + by + c = 0 and indicate the values of a, b and c in each case:

(i) 2x + 3y = 9.3̅5

We need to bring all terms to the left side to match the form ax + by + c = 0.
2x + 3y - 9.3̅5 = 0
Comparing this with ax + by + c = 0, we get:
a = 2
b = 3
c = -9.3̅5

(ii) x - y/5 - 10 = 0

This is already in the form ax + by + c = 0.
x - (1/5)y - 10 = 0
Comparing with ax + by + c = 0:
a = 1
b = -1/5
c = -10

(iii) -2x + 3y = 6

Subtract 6 from both sides:
-2x + 3y - 6 = 0
Comparing with ax + by + c = 0:
a = -2
b = 3
c = -6

(iv) x = 3y

Subtract 3y from both sides:
x - 3y = 0
This can be written as x - 3y + 0 = 0.
Comparing with ax + by + c = 0:
a = 1
b = -3
c = 0

(v) 2x = -5y

Add 5y to both sides:
2x + 5y = 0
This can be written as 2x + 5y + 0 = 0.
Comparing with ax + by + c = 0:
a = 2
b = 5
c = 0

(vi) 3x + 2 = 0

This equation has no y term, so we can write it as:
3x + 0y + 2 = 0
Comparing with ax + by + c = 0:
a = 3
b = 0
c = 2

(vii) y - 2 = 0

This equation has no x term, so we can write it as:
0x + y - 2 = 0
Comparing with ax + by + c = 0:
a = 0
b = 1
c = -2

(viii) 5 = 2x

Subtract 5 from both sides (or move 2x to the left):
2x - 5 = 0
This can be written as:
2x + 0y - 5 = 0
Comparing with ax + by + c = 0:
a = 2
b = 0
c = -5

Exercise 4.2

1. Which one of the following options is true, and why?
y = 3x + 5 has
(i) a unique solution,
(ii) only two solutions,
(iii) infinitely many solutions

Option (iii) infinitely many solutions is true.
Reason: A linear equation in two variables (like y = 3x + 5) represents a straight line. A line consists of infinite points, and every point on the line is a solution to the equation. For every value we choose for x, we get a corresponding unique value for y. Since we can choose infinite values for x, there are infinite corresponding values for y.

2. Write four solutions for each of the following equations:

(i) 2x + y = 7

We can express y in terms of x: y = 7 - 2x
1. Let x = 0, then y = 7 - 2(0) = 7. Solution: (0, 7)
2. Let x = 1, then y = 7 - 2(1) = 5. Solution: (1, 5)
3. Let x = 2, then y = 7 - 2(2) = 3. Solution: (2, 3)
4. Let x = 3, then y = 7 - 2(3) = 1. Solution: (3, 1)
The four solutions are (0, 7), (1, 5), (2, 3), and (3, 1).

(ii) πx + y = 9

Express y in terms of x: y = 9 - πx
1. Let x = 0, then y = 9 - π(0) = 9. Solution: (0, 9)
2. Let x = 1, then y = 9 - π(1) = 9 - π. Solution: (1, 9 - π)
3. Let x = -1, then y = 9 - π(-1) = 9 + π. Solution: (-1, 9 + π)
4. Let x = 2, then y = 9 - π(2) = 9 - 2π. Solution: (2, 9 - 2π)
The four solutions are (0, 9), (1, 9 - π), (-1, 9 + π), and (2, 9 - 2π).

(iii) x = 4y

Here it is easier to choose values for y and find x.
1. Let y = 0, then x = 4(0) = 0. Solution: (0, 0)
2. Let y = 1, then x = 4(1) = 4. Solution: (4, 1)
3. Let y = -1, then x = 4(-1) = -4. Solution: (-4, -1)
4. Let y = 2, then x = 4(2) = 8. Solution: (8, 2)
The four solutions are (0, 0), (4, 1), (-4, -1), and (8, 2).

3. Check which of the following are solutions of the equation x - 2y = 4 and which are not:

(i) (0, 2)

Substitute x = 0 and y = 2 in LHS:
LHS = 0 - 2(2) = -4
RHS = 4
Since LHS ≠ RHS, (0, 2) is not a solution.

(ii) (2, 0)

Substitute x = 2 and y = 0 in LHS:
LHS = 2 - 2(0) = 2
RHS = 4
Since LHS ≠ RHS, (2, 0) is not a solution.

(iii) (4, 0)

Substitute x = 4 and y = 0 in LHS:
LHS = 4 - 2(0) = 4
RHS = 4
Since LHS = RHS, (4, 0) is a solution.

(iv) (√2, 4√2)

Substitute x = √2 and y = 4√2 in LHS:
LHS = √2 - 2(4√2)
LHS = √2 - 8√2 = -7√2
RHS = 4
Since LHS ≠ RHS, (√2, 4√2) is not a solution.

(v) (1, 1)

Substitute x = 1 and y = 1 in LHS:
LHS = 1 - 2(1) = 1 - 2 = -1
RHS = 4
Since LHS ≠ RHS, (1, 1) is not a solution.

4. Find the value of k, if x = 2, y = 1 is a solution of the equation 2x + 3y = k.

Since x = 2 and y = 1 is a solution, these values must satisfy the equation.
Substitute x = 2 and y = 1 into the equation:
2(2) + 3(1) = k
4 + 3 = k
k = 7
Therefore, the value of k is 7.

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According to Edmund Halley, what is the principal use of the Analytic Art?

Answer

To bring mathematical problems to equations and exhibit them in the simplest terms possible.

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How many solutions does a linear equation in one variable have?

Answer

A unique solution (one and only one).

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Where is the solution of a linear equation in one variable typically represented visually?

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On a number line.

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What is the root of the equation $2x + 5 = 0$?

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$x = -\frac{5}{2}$

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How is the solution of a linear equation affected if the same number is added to both sides?

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The solution is not affected.

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The solution remains unchanged.

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In a cricket match, if two batsmen together scored 176 runs ($x$ and $y$), what is the representing linear equation?

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$x + y = 176$

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What is the general form of a linear equation in two variables?

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$ax + by + c = 0$

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In the standard form $ax + by + c = 0$, what types of numbers are $a$, $b$, and $c$?

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Real numbers.

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What is the specific condition for $a$ and $b$ in the linear equation $ax + by + c = 0$?

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$a$ and $b$ are not both zero.

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Express the equation $2x + 3y = 4.37$ in the form $ax + by + c = 0$.

Answer

$2x + 3y - 4.37 = 0$

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In the equation $x - 4 = \sqrt{3}y$, what is the value of $b$ when written in standard form?

Answer

$b = -\sqrt{3}$

Question

Identify the value of $c$ for the equation $4 = 5x - 3y$ when written in the form $ax + by + c = 0$ where $a=5$.

Answer

$c = -4$

Question

What is the value of $c$ in the linear equation $2x = y$ when expressed in standard form?

Answer

$c = 0$

Question

How can the linear equation in one variable $ax + b = 0$ be written as a linear equation in two variables $x$ and $y$?

Answer

$ax + 0y + b = 0$

Question

Express $x = -5$ as a linear equation in two variables $x$ and $y$.

Answer

$1x + 0y + 5 = 0$

Question

Express $y = 2$ as a linear equation in two variables $x$ and $y$.

Answer

$0x + 1y - 2 = 0$

Question

Write $2x = 3$ as a linear equation in two variables $x$ and $y$ in standard form.

Answer

$2x + 0y - 3 = 0$

Question

The cost of a notebook ($x$) is twice the cost of a pen ($y$). Write this as a linear equation in two variables.

Answer

$x = 2y$ (or $x - 2y = 0$)

Question

Identify the value of $b$ in the equation $x - \frac{y}{5} - 10 = 0$.

Answer

$b = -\frac{1}{5}$

Question

What is the value of $a$ in the equation $3x + 2 = 0$ when expressed in two variables $x$ and $y$?

Answer

$a = 3$

Question

What is the value of $a$ in the equation $y - 2 = 0$ when expressed as a linear equation in two variables $x$ and $y$?

Answer

$a = 0$

Question

Express the equation $5 = 2x$ in the standard form $ax + by + c = 0$.

Answer

$2x + 0y - 5 = 0$

Question

What constitutes a 'solution' for a linear equation involving two variables?

Answer

A pair of values, one for $x$ and one for $y$, which satisfy the equation.

Question

How is a solution for a linear equation in two variables written as a coordinate?

Answer

As an ordered pair $(x, y)$.

Question

Why is $(3, 2)$ a solution to the equation $2x + 3y = 12$?

Answer

Substituting $x=3$ and $y=2$ results in $2(3) + 3(2) = 12$, which satisfies the equation.

Question

In an ordered pair solution $(x, y)$, which variable's value is written first?

Answer

The value for $x$.

Question

How many solutions exist for a linear equation in two variables?

Answer

Infinitely many solutions.

Question

If $x=2$ is chosen for the equation $2x + 3y = 12$, what is the corresponding value of $y$?

Answer

$y = \frac{8}{3}$

Question

Find one solution for $x + 2y = 6$ by setting $x = 0$.

Answer

$(0, 3)$

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Find one solution for $x + 2y = 6$ by setting $y = 0$.

Answer

$(6, 0)$

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What is an easy way to find a specific solution for a linear equation in two variables?

Answer

Set one variable (like $x$) to zero and solve for the other variable ($y$).

Question

Find a solution for $4x + 3y = 12$ where $y = 0$.

Answer

$(3, 0)$

Question

Identify the common solution $(x, y)$ that satisfies the equation $2x + 5y = 0$ when $x = 0$.

Answer

$(0, 0)$

Question

For the equation $3y + 4 = 0$, what is the value of $y$ for any value of $x$?

Answer

$y = -\frac{4}{3}$

Question

Which option is true for the equation $y = 3x + 5$: (i) unique solution, (ii) two solutions, or (iii) infinitely many solutions?

Answer

(iii) infinitely many solutions

Question

Find a solution for $2x + y = 7$ when $x = 0$.

Answer

$(0, 7)$

Question

Find a solution for $\pi x + y = 9$ when $x = 0$.

Answer

$(0, 9)$

Question

Find a solution for the equation $x = 4y$ other than the origin.

Answer

Any pair where $x$ is four times $y$, such as $(4, 1)$.

Question

Is the ordered pair $(0, 2)$ a solution for the equation $x - 2y = 4$?

Answer

No ($0 - 2(2) = -4$, not $4$).

Question

Is the ordered pair $(4, 0)$ a solution for the equation $x - 2y = 4$?

Answer

Yes ($4 - 2(0) = 4$).

Question

Is the ordered pair $(1, 1)$ a solution for the equation $x - 2y = 4$?

Answer

No ($1 - 2(1) = -1$, not $4$).

Question

Determine the value of $k$ if $x = 2$ and $y = 1$ is a solution of $2x + 3y = k$.

Answer

$k = 7$

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True or False: Every point on the graph of a linear equation in two variables is a solution of that equation.

Answer

True.

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True or False: Every solution of a linear equation in two variables is a point on its graph.

Answer

True.

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Where do the solutions of a linear equation in two variables appear when plotted?

Answer

On the Cartesian plane.

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In the equation $3x + 0y + 2 = 0$, what is the value of $b$?

Answer

$0$

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In the equation $0x + 1y - 2 = 0$, what is the value of $a$?

Answer

$0$

Question

What is the coefficient of $y$ in the equation $x = 3y$ when written in standard form?

Answer

$-3$

Question

Express $2x = -5y$ in the form $ax + by + c = 0$.

Answer

$2x + 5y + 0 = 0$

Question

Identify the variables used in the linear equation example $1.2s + 3t = 5$.

Answer

$s$ and $t$

Question

In the equation $\pi u + 5v = 9$, what is the value of the coefficient $a$?

Answer

$\pi$

Question

When solving $2x + 3y = 12$, if we choose $x = -5$, the equation reduces to what linear equation in one variable?

Answer

$-10 + 3y = 12$

Question

In the equation $4 - 3x = 0$, how is this expressed with two variables $x$ and $y$?

Answer

$-3x + 0y + 4 = 0$

Question

What is the value of $y$ in the ordered pair solution $(\sqrt{2}, 4\sqrt{2})$ for the equation $x - 2y = 4$?

Answer

$4\sqrt{2}$

Question

Why is the solution $(0, 4)$ different from the solution $(4, 0)$ for a linear equation?

Answer

Because the values are assigned to different variables ($x$ vs $y$) in the ordered pair.