Linear Graphs: The information gathered might be grouped in a sequence or a frequency distribution table first. The information from a frequency distribution table might be represented visually in the form of graphs.

A cartesian plane is known to be divided into four quadrants. We also know that a line graph is made up of successively linked line segments. At times, the graph consists of a single straight line. This type of graph is known as a linear graph. To draw a linear graph, we must first mark several points on graph paper.

In this article, we will explore and learn about linear graphs in detail.

Drawing a Linear Graph

When we plot points in the Cartesian plane, we sometimes get a straight line; when the points lie on a straight line, we get a linear graph. In our everyday life, we deal with many quantities in which one value affects the other. For example, the quantity of petrol purchased will affect the amount to be paid; similarly, the side of a square will affect the perimeter and the area of the square.

Let us see how these graphs can be helpful to us in evaluating the values of the dependent and independent variables.

Independent Variable: The variable whose value does not depend on the value of other variables is known as the independent variable.

Dependent Variable: The variable whose value depends upon the value of other variables is known as the dependent variable.

The general equation to represent a straight line is shown below:

Let us understand this with the help of an illustration.

A vehicle is moving at a constant speed of \(80 \mathrm{~km} / \mathrm{hr}\). Draw a distance-time graph to represent the same and then use the graph to find the distance covered by the vehicle in \(4\) hours \(30\) minutes.

We are aware of the formula that distance \( = {\rm{speed}} \times {\rm{time}}\)

Now, the vehicle is moving at a speed of \(80 \mathrm{~km} / \mathrm{hr}\); therefore, the distance covered by the vehicle in time \((t)\) is equal to \(80\,\text {t}\).

Now, let us make the table.

Time (in Hours)	\(1\)	\(2\)	\(3\)	\(4\)	\(5\)
Distance Covered (in \(\text {km}\))	\(80\)	\(160\)	\(240\)	\(320\)	\(400\)

Now next task is to choose a suitable scale for both the axis.

On \(x\) – axis, we choose the scale as \(1\) unit \(=1\) hour.

On \(y\) – axis, we choose the scale as \(1\) unit \(=80 \mathrm{~km}\).

We plot points \((1,80),(2,160),(3,240),(4,320)\), and \((5,400)\) on the graph paper.

The graph obtained is the linear graph.

Now, we have to find the distance covered by the vehicle in \(4\) hours \(30\) minutes.

If we look at the graph, corresponding to \(4\) hours \(30\) minutes or \(4 \frac{1}{2}\) hours on the horizontal axis at \(L\), we get \(360 \mathrm{~km}\) on the vertical axis at point \(M\). Thus, distance covered by the vehicle in \(4 \frac{1}{2}\) hours is \(360 \mathrm{~km}\).

Graphs of Linear Equations in Two Variables

The graph of a linear equation in two variables is always a straight line. To draw the graphs of linear equations in two variables \(x\) and \(y\), proceed as follows:

1. Rewrite the given equation with \(y\) as a subject.
2. Select at least three convenient values of \(x\) and find the corresponding values of \(y\) for each of the selected values of \(x\).
3. Make a table of the values.
4. Draw the axes on the graph and then choose the suitable scale.
5. Plot the points on the graph paper.
6. Connect any two points by a straight line and check that the third point lies on it. In fact, two points are sufficient to mark the position of a straight line, but we use at least three points for drawing the graph to check the correctness of the graph.

Let us understand the above points with the help of an illustration.

Draw the graph of \(3 x-2 y=2\).

The given equation is \(3 x-2 y=2\), it can be written as \(2 y=3 x-2\) or \(y=\frac{3}{2} x-1 \ldots…(1)\)

Now, the next step is to select the appropriate values of \(x\), say, \(0,2,4\) and find the corresponding values of \(y\) by using equation \((1)\).

Thus, when \(x=0, y=\frac{3}{2} \times 0-1=-1\)

\(x=2, y=\frac{3}{2} \times 2-1=3-1=2\)

\(x=4, y=\frac{3}{2} \times 4-1=6-1=5\)

Thus, the table becomes:

\(x\)	\(0\)	\(2\)	\(4\)
\(y\)	\(-1\)	\(2\)	\(5\)

Study The Graph Of Linear Equation

Select coordinate axes and take \(1 \mathrm{~cm}=1\) unit on both the axes.

Plot the points \((0,-1),(2,2)\) and \((4,5)\) on the graph paper. Connect any two points by a straight line. The graph of the given linear equation is shown below.

Solved Examples – Linear Graphs

Q.1. Draw a graph to convert miles to kilometres given that \(1\) mile \(=1.6 \mathrm{~km}\) approximately. Use the graph to find how many miles are approximately equal to \(7 \mathrm{~km}\).
Ans: Let us consider \(x\) – axis to be miles and \(y\) – axis to be kilometres. Also, \(x=1\) mile, then \(y=1.6 \mathrm{~km}\).
We find that kilometre \((y)\) is dependent upon miles \((x)\). For different values of \(x\) and \(y\), let us form a table.

Miles \((x)\)	\(0\)	\(1\)	\(2\)	\(3\)	\(4\)	\(5\)
Kilometres \((y)\)	\(0\)	\(1.6\)	\(3.2\)	\(4.8\)	\(6.4\)	\(8\)

We will take the miles on the \(x\) – axis and kilometres along the \(y\) – axis.

Thus, plot the points \((0,0),(1,1.6),(2,3.2),(3,4.8),(4,6.4)\) and \((5,8)\). Join all the plotted points together, and we will observe that we get a straight line as shown in the graph.

Therefore, we get a linear graph \(A B C D E\) showing a linear relationship between miles and kilometres. To find the miles that are equal to \(7 \mathrm{~km}\), draw a horizontal line from \(7 \mathrm{~km}\), which intersects the line at \(Q\). From \(Q\), draw a vertical line that intersects the horizontal axis at \(4.4\) miles.
Therefore, \(7 \mathrm{~km}\) is approximately equal to \(4.4\) miles.

Q.2. The graph given below shows the change in the temperature of a block when heated. Use the graph to answer the following questions.
a) For how many seconds the block had no change in temperature?
b) What was the temperature when \(t=30\) seconds?
c) How long was the block in a liquid state?

Ans: After observing the graph.
a) For \(20\) seconds, the block had no change in temperature.
b) The temperature was \(40^{\circ} \text {C}\) when \(t=30\) seconds.
c) As we can see, the block started melting after \(20\) seconds, and after \(50\) seconds, the water started boiling. Thus, the block was in a liquid state for \((50-20)\) seconds \(=30\) seconds.
Hence, \(30\) seconds is the required answer.

Q.3. Draw the graph of \(2 x=-y+1\)
Ans: The given equation is \(2 x=-y+1\)
The equation can also be written as \(x=\frac{-y}{2}+\frac{1}{2}\)
Thus, the table of the values will be:

\(x\)	\(0\)	\(1\)	\(-1\)
\(y\)	\(1\)	\(-1\)	\(3\)

Plot the points \((0,1),(1,-1)\) and \((-1,3)\) on the graph paper. Connect any two points by a straight line.
Therefore, the graph of the given equation is shown below.

Q.4. The graph of a linear equation in \(x\) and \(y\) passes through \(A(-1,-1)\) and \(B(2,5)\). Find the value of \(h\) and \(k\) if the graph passes through \((h, 4)\) and \(\left(\frac{1}{2}, k\right)\)
Ans: Take the scale as \(1 \mathrm{~cm}=1\) unit on both axes. Plot the points \(A(-1,-1)\) and \(B(2,5)\) on the graph paper and draw a straight line passing through these points.

Through \(y=4\), draw a horizontal line to meet the graph of the straight line \(A B\) at the point \(P\). Through \(\mathrm{P}\), draw the vertical line which meets \(x\) – axis at \(x=\frac{3}{2}\).
l.e., \(h=\frac{3}{2}\)
Similarly, through \(x=\frac{1}{2}\), draw a vertical line to meet the graph of the straight \(A B\) at \(Q\). Through \(Q\), draw the horizontal line which meets \(y\) – axis at \(y=2\), i.e., \(k=2\)
Hence, \(h=\frac{3}{2}\) and \(k=2\).

Q.5. Draw the line passing through \(A(2,5)\) and \(B(-4,-5)\). Find the coordinates of the point at which this line meets the \(x\) – axis.
Ans: Draw \(x\) – axis and \(y\) – axis and mark points on them using a convenient scale. Plot the \(A(2,5)\) and \(B(-4,-5)\). Join \(A\) and \(B\), and produce \(A B\) in both directions. We get the linear graph.

The line \(A B\) meets the \(x\) – axis at \((-1,0)\)

Summary

In this article, we learnt about graphs, especially linear graphs. We learnt that linear graphs are the graphs when there is a single straight line. In addition to this, we also learnt about the graphs of linear equations in two variables. To master the concepts, we solved some more examples related to linear graphs.

Frequently Asked Questions (FAQs) – Linear Graphs

Q.1. What is a linear graph?
Ans: When we plot points in the Cartesian plane, we sometimes get a straight line; when the points lie on a straight line, we get a graph known as a linear graph.

Q.2. What is an example of a linear graph?
Ans: The example of a linear graph is the quantity of petrol purchased will affect the amount to be paid; similarly, the side of a square will affect the perimeter and the area of the square.

Q.3. How do you find a linear equation from a graph?
Ans: The general form to find a linear equation is \(y=m x+c\), where \(x\) and \(y\) are variables, and \(m\) is slope or gradient.

Q.4. How does a graph of a linear equation in two variables look like?
Ans: The graph of a linear equation in two variables looks like a straight line.

Q.5. How do you draw linear graphs?
Ans: To draw a linear graph, we need to locate some points on the graph. Then, we have to join those points. If the line joining these points is a straight line, then we get a linear graph.

Learn About The Solution Of A Linear Equation

We hope this detailed article on Linear Graphs has helped you in clearing your doubts. If you have any issues do let us know about it in the comments section below and we will get back to you soon.