Chapter-4 Linear Equations in Two Variables

Explanation: 

Assuming 2x - 5y = 7,

By switching out 5y = 2x - 7

The result is y = (2x - 7)/ 5.

Here, we shall obtain various y values for various x values.

As a result, there are innumerable solutions to the linear equation.

Explanation: 

The formula is 2x + 5y = 7.

Let's think about the value (1, 1).

a. (1, 1) is the answer to 2x + 5y = 7.

b. If positive real numbers are taken into account, the equation 2x + 5y = 7 will have a lot of solutions.

c. If real numbers are taken into account, 2x + 5y = 7 will have an unlimited number of answers.

If rational numbers are taken into account, there are multiple solutions to the equation 2x + 5y = 7.

We are aware that a two-variable linear equation has an infinite number of solutions.

There is a collection of infinite solution points for the linear equations system where an equation's L.H.S changes to an R.H.S.

Therefore, if x and y are natural numbers, the equation has a singular solution.

Explanation: 

2x + 3y = k is the given linear equation.

Here, the point is given as (x, y) = (2, 0).

Let's change the values in the equation 2 (2) + 3(0) = k that are given.

Calculating further, 4 + 0 = k

So, we obtain k = 4.

Consequently, k has a value of 4.

Explanation: 

2x + 0y + 9 = 0 is the linear equation that is given.

It can be expressed as 2x + 9 = 0.

2x = -9

 x = -9/2

Since the y coefficient is 0, any value can be used with no effect on the solution.

As a result, the two-variable linear equation has the form (-9/2, m).

Explanation: 

2x + 3y = 6 is the given linear equation.

Given that the equation crosses the y-axis, the x-coordinate is zero.

In the linear equation 2 (0) + 3y = 6,

x = 0 is substituted.

0 + 3y = 6

3y = 6

3 y divided by both sides yields 2

therefore (0, 2) are the coordinates.

As a result, the point (0, 2) on the graph of the linear equation is where the y-axis is broken.

Explanation: 

The response is B.

Here, the coefficient of 'y' in the formula x = 7 is equal to zero.

As a result, the formula is 1.x + 0.y = 7.

The necessary equation is therefore 1.x + 0.y = 7.

Explanation: 

The response is C.

The y-coordinate of any point on the X-axis is equal to zero, or y = 0.

The generic form of each point on the X-axis is hence (x, 0).

Explanation: 

Any point on the line y = x is known to have the same x and y coordinates.

Therefore, any point on the line y = x will be (a, a). Any point on the line y = x will therefore be (a, a).

Explanation: 

The response is B.

All the X-axis points have the same y-coordinate, which is 0.

As a result, the X-axis equation has the form y = 0.

Explanation: 

y = 6 is the given equation.

Y = 0 can be used to express it.x + 6

Here, y equals 6 for every value of x.

The points are 0, 6, 1, 6, 3, and 6...

These points can be plotted to produce a straight line parallel to the x-axis at a distance of 6 units from the x-axis.

The line is therefore 6 units from the origin and parallel to the x-axis.

Explanation: 

In (a) x+2y=7, substituting the values x = 5 and y = 2 yields LHS=5+22=5+4=9#7=RHSi.e.LHSRHS in 

(b) 5x+2y=7, LHS=5+2=7=RHSi.e.LHS=RHS in

(c), and LHS=5+2=27=RHSi.e.LHSRHS in

Therefore, (c) is the appropriate choice.

Explanation: 

The right answer is A x + y = 0.

For (-2, 2):

 -2 + 2 = 0

For (0, 0):

 0 + 0 = 0

For (2, -2):

 2 + (-2) = 0

Thus, the equation for which the provided points represent the answer is x + y = 0.

Explanation: 

Think of the favorable answers as (p, q).

Since the abscissa, p, is positive, it will be located along the x-axis' positive direction.

Given that the ordinate, q, is positive, it will lie towards the y-axis's positive direction.

Consequently, (p, q) will be in the first quadrant.

As a result, the first quadrant will contain the positive solutions.

Explanation: 

A linear equation is one that has a degree of 1 as its maximum value.

In a linear equation, this means that no variable's exponent can be greater than 1.

2x + 3y = 6 is the given linear equation.

It intersects the x-axis, indicating that y = 0.

Let's replace it in the formula 2x + 3(0) = 6

2x = 6

2 x divided by both sides yields 3

As a result, the linear equation's graph is (3, 0).

Explanation: 

As far as we are aware, a linear equation is one in which the highest power of the variable is consistently 1. A linear equation is one that has a degree of 1 as its maximum value.

The x and y coordinates in the linear equation y = x are identical.

The graph comes from the suggested choice (1, 1).

Consequently, (1, 1) is where the linear equation's graph intersects.

Explanation: 

Take the equation x + 2y = 5 as an example. (1)

In the equation 1 + 2y = 5 2y = 5 - 1 2y = 4, replace x = 1

two sides divided by two equals two.

The answer to the problem is (1, 2)

Multiplying equation (1) by three yields 3x + 6y = 15.

Replace with x = 1 and y = 2

LHS = 3(1) + 6(2) = 3+12=15, which equals RHS.

By a non-zero value multiplied by an equation, the answer remains the same.

Equation (1) is now divided by 2 by x/2 + y = 5/2.

Change equation (1) to read x = 1 and y = 2.

LHS = 1/2 + 2

LCM = (1 + 4)/2 = 5/2 = RHS after taking

By using a non-zero number to divide an equation, the result remains the same.

Consequently, the answer is still the same.

Explanation: 

The response is C.

Let axe by by and c equal zero in the linear equation.

When we change x to 1 and y to 2, we get the following equation: => a + 2b + c = 0, where a, b, and c are real numbers.

Here, a+ 2b + c = 0 is satisfied for various values of a, b, and c.

Consequently, x = 1 and y = 2 can satisfy an endless number of linear equations in x and y.

Explanation: 

The provided point has the form (a, a).

Here, the values of the x and y coordinates are identical.

It must therefore be on the line y = x.

The point is therefore on the line y = x.

Explanation: 

D x + y = 0 x + y = a + (- a) = 0 is the right answer because the provided point has the form (a, -a).

Since x + y = 0, the point (a, -a) is always on that line.

Explanation: 

Given is the linear equation 3x + 4y = 12.

The provided point is (0, 3).

In the equation 3 (0) + 4 (3) = 12 12 = 12, let's substitute it.

The assertion is accurate as a result.

Explanation: 

Given is the linear equation x + 2y = 7.

The indicated point is (0, 7).

Let's replace it in the formula 0 + 2 (7) = 7 14 7

Explanation: 

Given is the linear equation x + y = 0.

It can be written as

x = -y

The graph's points are (-3, 3) and (-1, 1)

Taking into account (-3, 3) x = -3 and y = 3 -3 = 3

fulfils the equation

In light of (-1, 1)

x = -1 and y = 1 -1 = 1

fulfils the equation

The assertion is accurate as a result.

Explanation: 

based on the graph

At a distance of 3 units to the right of the origin, a line is perpendicular to the y-axis.

Consequently, x = 3 will be the linear equation.

The assertion is accurate as a result.

Explanation: 

The given equation, xy+2=0, can be satisfied by the points (0,2), (1,3), (2,4), and (4,6).

The equation xy+2o is solved at each of these positions.

However, if we substitute (3,-5), i.e. 3(-5)+2=0, i.e. 3+5=0, or 10=0, it does not meet the equation and is therefore untrue.

Because the point (3, 5) does not meet the provided equation, the stated statement is untrue.

Explanation: 

The dots that are the answers to the given problem are connected to form a graph of a linear equation in two variables.

A solution to the linear equation must therefore exist at every point on the graph.

As a result, the assertion is untrue.

Explanation: 

Every two-variable linear equation has the form y=mx+c. 

In this case, y has a distinct value for each value of x. 

Additionally, x and y are directly related.

As a result, xy will never change.

As a result, the graph will continue to have a distinct slope, which causes it to be a straight line. The assertion is untrue. 

Explanation:
Given, the equations in linear form are y = x and y = -x.

The equations' graphs must be drawn on the same cartesian plane.

To plot a few points on the graph using the equation y = x, we require a few data points.

For x = 1, y = 1

For x = 4, y = 4

The points of the equation are (1, 1) and (4, 4), respectively.

The graph of y = x is created by connecting the points (1, 1) and (4, 4) on the graph to form a straight line.

We need a few points to plot on the graph when y = -x is taken into consideration.

For x = 3, y = -3

For x = -4, y = 4

As a result, the equation's points are (3, -3) and (-4,4) The graph of y = -x is obtained by combining the points (3, -3) and (-4, 4), which are plotted on the graph as the points (3, -3) and (-4, 4) respectively.

We can see that the graphs of y = x and y = -x overlap at the point (0, 0) in both cases.

Explanation: 

Given equation is 2x + 5y = 19, which is (1)

Considering that the ordinate is 1.5 times the abscissa

Therefore, y = 3x /2

Hence equation (1) becomes 2x + 5(3x /2) = 19 19x/2 = 19

x thus equals 2

y = 3x/2 = (3 × 2)/2 = 3

The graph's point is therefore (2, 3).

Explanation: 

The equation y = - k, where k is the distance of the line from the x-axis, can be used to calculate any straight line that is perpendicular to the X-axis and below the X axis.

Here, k = 3

The line's equation will therefore be y=-3.

The needed graph is this one.

Explanation: 

The graph's coordinates add up to 10 units, as is obvious.

The linear equation, whose solutions are represented by points with coordinates that add up to 10, must be graphed.

Let the coordinates be x and y.

Assuming x + y = 10,

When x=0

 0+y=10,

y=10

Y = 10 - 3

y = 7 when x = 3 and

3 + y = 10.

When y = 10 - 5 and x = 5, the result is y = 5.

When x = 10 and 10 + 10 + 10 - 10 y = 0, y = 0.

The points are as follows: (0, 10), (3, 7), (5, 5) and (10, 0).

The graph of the equation x + y = 10 is obtained by plotting the points on the graph.

Explanation: 

Given that the ordinate is three times the abscissa

The graph's linear equation must be written down.

Times the x-coordinate equals the y-coordinate.

So, y = 3x

If x = 1, then (1) y = 3

Y = 3(2) y = 6 when x = 2.

The points are therefore (1, 3), and (2, 6).

Plotting the graph's points (1, 3), and (2, 6),

We obtain a straight line AB by connecting the points, and its equation is y = 3x.

Consequently, y = 3x is the linear equation.

Explanation: 

Given: The equation 3y = ax + 7 is linear.

By changing the values of x and y in the given equation, we may determine the value of 'a'.

We get 3(4) = a(3) + 7 by changing the values of x = 3 and y = 4 in the equation 3y = axe + 7.

12 = 3a + 7

3a = 5 a = 5/3

Consequently, a = 5/3.

Explanation: 

Given: The equation 3y = ax + 7 is linear.

By changing the values of x and y in the given equation, we may determine the value of 'a'.

We get 3(4) = a(3) + 7 by changing the values of x = 3 and y = 4 in the equation 3y = axe + 7.

12 = 3a + 7

3a = 5 a = 5/3

Consequently, a = 5/3.

33. How many solution(s) of the equation 2x + 1 = x – 3 are there on the : (i) Number line (ii) Cartesian plane 

Explanation: 

The preceding equation can be represented as 1.x + 0.y = - 4 and is 2x + 1 = x - 3 2x - x = -3 - 1 x = -4....(ii)

(i) On the x-axis, a number line represents each real value of x. As a result, the number line's point at x = -4 is exactly one on the line.

(ii) In contrast, x + 4 = 0 depicts a straight line that is perpendicular to the y-axis. On a line in the Cartesian plane, there are an infinite number of points.

Explanation: 

The formula is x + 2y = 8.(i)

(i) Enter y = 0 in Eq. when the point is on the x-axis. (i). We discover that x + 2 0 = 8 x = 8 and that the necessary point is at (8, 0).

(ii) Enter x = 0 in Eq(i) when the point is on the y-axis. We discover that 

0 + 2y = 8

y=82=4

The necessary point is therefore (0, 4).

Explanation: 

Given that 2x + cy = 8 is a linear equation.

The solution to the equation has equal values for x and y.

The value of c must be determined.

Assuming x = y

In the preceding equation, substitute y = x, 2x + c(x) = 8, 

cx = 8 - 2x,

and c = (8 - 2x)/x.

where x is greater than zero.

Consequently, the formula for c is (8 - 2x)/x.

Explanation: 

As x changes, y does also.

Y equals 12 when x equals 4.

In order to determine the value of y for x = 5, we must write a linear equation.

Since y is directly related to X, Y ∝ X

Consequently, y = kx is the linear equation.

where k is a randomly chosen constant.

From the starting point, 12 = k(4),

k = 12/4,

 k = 3.

Consequently, y = 3x is the linear equation.

In the equation, enter x = 5,

y = 3(5),

 y = 15.

Consequently, y has a value of 15.

Explanation: 

First, create the equation's graph.

y = 9x - 7

When x = 2 and y = 9, the linear equation at this point is: 2 - 7 = 18 - 7 = 11.

Y = 9 -2 - 7 when x = -2.

= -18 - 7 = -25

 x2−2y11−25

Two points are located here: D(2, 11) and E(-2, -25).

In order to create the graph, plot the points and connect D and E.

On the graph paper, we now plot the provided points A(1, 2), B(-1, -16), and C(0, -7). We can see that every point is on the DE line.

Explanation: 

The points (6, -2) and (-6, 6) are given.

The positions (6, -2) and (-6, 6) should satisfy the linear equation y = mx + c.

At the location (6, -2), -2 = 6m + c.(i)

At point (-6, 6), 6 = -6m + c, as well.(ii) We obtain, after solving equations (i) and (ii),

replacing c = 2 in equation (i)

  6m+c=−2

−6m+c=   6

________________

    0+2c=  4

________________

⇒[c=2]

⇒6m+2=−2 

⇒6m=−2−2 

⇒6m=−4 

⇒m=−46

 ⇒[m=−23]

In the linear equation y=mx+c, we get m=23 and c=2.

⇒y=−23x+2⇒y=−2x+63⇒3y=−2x+6

When the linear equation's graph,

Cuts the x-axis (i)

Put y = 0 in equation 2x + 3y = 6

to get 2x = 6 and x = 3.

When the linear equation's graph,

Cuts the y-axis (ii)

Add x = 0 to the equation 2x + 3y = 6 to obtain 2.0+3y=6 3y=6 y=2

As a result, the x-axis and y-axis of the graph of the linear equation are split at the points (3, 0) and (0, 2), respectively.

Explanation: 

Given that 3x + 4y = 6 is a linear equation.

The x-axis and the y-axis must be intersected at specific points on the equation's graph, which must be drawn.

You may rewrite the equation as 4y = 6 - 3x

y = (6 - 3x)/4.

If x = 0, then 

y = (6 - 3(0)) / 4

and y = 6/4

and y = 3/2.

If y = 0, then 0 = (6 - 3x)/4 

6 - 3x = 0

and 3x = 6

x = 6/3

x = 2

The points are therefore (2, 0) and (0, 3/2)

When the points are plotted on the graph,

We create a straight line by connecting the points, which is the graph of the equation 3x + 4y = 6.

The graph's x-axis is broken at the point (2, 0), as can be seen.

The graph's y-axis is broken at (0, 3/2)

Explanation: 

Given that C = (5F - 160)/9 may be used to convert between Fahrenheit (F) and Celsius (C),

C = F given

We must determine the temperature's numerical value.

You can rewrite the relationship as 9C = 5F - 160.

In the provided relation, 9F = 5F - 160, replace C = F.

9F - 5F = -160

4F = -160

 F = -160/4 

F = -40

As a result, the temperature is represented by the number 40.

Explanation: 

Given that a liquid's temperature is x°K in Kelvin units.

A liquid's temperature is expressed as y°F in Fahrenheit units.

The equation y = (9/5)(x - 273) + 32 describes the relationship between the two temperature measuring systems.

If the temperature is 158° F, we must determine the temperature in Kelvin.

Given that y=158° F

When the value of y is substituted in the given relationship, 158 = (9/5)(x - 273) + 32

158 - 32 = (9/5)(x - 273)

14(5) x - 273 = 70 

x = 70 + 273 

x = 343°K where 126(5/9) = x - 273 x - 273

The temperature is therefore 343°K.

Explanation: 

Given that the acceleration caused by pulling a cart is directly proportional to the force applied, this makes sense.

By using the constant mass of 6 kg, we must build a linear equation and graph its solution.

The force needed when the acceleration is 5 m/sec2 must be deduced from the graph.

A given, F

So, F = ma

Where m, or constant mass, is an arbitrary constant.

Assuming m = 6 kg

F = 6a (i) now. When the acceleration is 5 m/sec2, the force needed is F = 6(5).

F = 30 N

The force is 30 N (i) as a result. When the acceleration is 6 m/sec2, the force required is F = 6(6) F = 36 N.

The force is therefore 36 N.

The rating is (5,

The graph of the equation F = 6a is a straight line that we are able to produce.