** **In this mini-lesson, we will explore solving a system of graphing linear equations using different methods, linear equations in two variables, linear equations in one variable, solved examples, and pair of linear equations.

For example, \(2x-5y+21=0\) is a linear equation.

But how do you represent linear equations on a graph?

To plot a graph of linear equations in two variables, we have different methods.

Let us learn in detail about them in the following lesson.

**Lesson Plan**

**What Do You Mean by Graphing Linear Equations?**

Linear equations are algebraic equations in which each term has a real constant and the equation contains 2 variables of highest power 1.

We represent the linear equation in \(y=mx+b\) form, also known as the y-intercept form.

The representation of a linear equation on a graph is called a graphing linear equations in two variables.

Also read:

What are Linear Equations in One Variable?-Olympiad

Let us see an example of graphing a linear equation:

We have to represent the equation \(x+2y=7\) in graph.

Here, the equation \(x+2y=7\) makes a straight line on the graph.

Similarly, all linear equations create a straight line on the graph.

**Note:** X-intercept is the point where any line crosses the x-axis on the graph.

Y-intercept is the point where any line crosses the y-axis on the graph.

**How Do You Represent Linear Equation on Graph?**

- Make sure the linear equation is in y-intercept form, which is \(y = mx + b\).
- Apply the trial and error method and find the value of (x, y) up to three pairs, which satisfy the linear equation.
- Find the x-intercept and y-intercept of the equation.

For y-intercept, substitute the value of x = 0 in the equation. This results in x = a.

For x-intercept, substitute the value of y = 0 in the equation. This results in y = c. - Thus, the points are (a, 0) and (0, c). Make a tabular form and put the value of x and y respectively.
- Plot all the points on the graph paper.
- Join all the points which are marked on the graph paper and get a straight line that represents the given linear equation graphically.

Let's take an example:

**Example**

Draw a graph of the linear equation \(x+2y=7\).

**Solution**

We have to convert the given linear equation \(x+2y=7\) in the form of y = mx + b.

On converting, we get: \(x=7-2y\)

We need to find the x and y-intercept respectively.

For that, put y=0 in the equation.

\(x=7-2(0)\)

\(x=7\)

Put x=0 in the equation.

\(2y=7-(0)\)

\(y=\dfrac72 = 3.5\)

Now, we will apply the trial and error method and find 3 pairs of values of (x, y) which satisfy the given linear equation \(x=7-2y\)

See the values of x and y in the following table.

Plot the points (7,0), (5,1), and (3,2) on the graph.

Join all the points which are marked on the graph paper and get a straight line that represents the given linear equation graphically.

- Linear equation in two variables has infinitely many solutions.
- A graph of linear equation is always a straight line.
- The equation y = mx is always passing through the origin (0, 0).

**Solved Examples**

Example 1 |

Isabella has the linear equation x - 2y = 2. Help her to draw the linear equation on the graph.

**Solution**

The given linear equation is \(x-2y=2\). Convert the equation in the form of y = mx + b.

\[y=\dfrac x2 - 1\]

We need to find the x and y-intercept respectively.

For that, put y=0 in the equation.

\[x=2(0)+2\]

\[x=2\]

Put x=0 in the equation.

\[2y=(0) - 2\]

\(y=1\)

Now, we will apply the trial and error method, and find 3 pairs of values of (x, y) which satisfy the given linear equation \(y=\dfrac x2 - 1\).

See the values of x and y in the following table.

Plot the points (2,0),(4,1),(0,-1) on the graph.

Join all the points which are marked on the graph paper and get a straight line that represents the given linear equation graphically.

Example 2 |

William wants to plot the graph of \( - \frac{1}{2}x + \frac{1}{3}y = 1\). Help him to plot the graph for linear equation.

**Solution**

We determine any two specific points for the solution of the equation:

\[x = \] |
\(2\) |
\(0\) |

\[y = \] |
\(6\) |
\(3\) |

Solution: |
\[\left( {2,6}\right)\] |
\[\left( {0,3}\right)\] |

The graph of the equation is plotted below:

- The sum of the digits of a two-digit number is 8. When the digits are reversed, the number is increased by 18. Find the number.
- Jake's piggy bank has 11 coins (only quarters or dimes) that have a value of $1.85. How many dimes and quarters does the piggy bank have?

**Interactive Questions**

**Here are a few activities for you to practice. **

**Select/type your answer and click the "Check Answer" button to see the result.**

**Let's Summarize**

The mini-lesson targeted the fascinating concept of graphing linear equations. The math journey around graphing linear equations starts with what a student already knows, and goes on to creatively crafting a fresh concept in the young minds. Done in a way that is not only relatable and easy to grasp, but will also stay with them forever. Here lies the magic with Cuemath.

**About Cuemath**

At Cuemath, our team of math experts is dedicated to making learning fun for our favorite readers, the students!

Through an interactive and engaging learning-teaching-learning approach, the teachers explore all angles of a topic.

Be it worksheets, online classes, doubt sessions, or any other form of relation, it’s the logical thinking and smart learning approach that we, at Cuemath, believe in.

**Frequently Asked Questions (FAQs)**

## 1.How do you graph a linear equation?

The basic methods of the graphing linear equation:

- The first method is by plotting all points on the graph and then drawing a line through the points.
- The second is by using the y-intercept of the equation and slope of the equation.

## 2.What are the 3 ways to graph a linear equation?

The 3 ways to graph a linear equation:

- Using two points to plot the graph of a linear equation.
- Use the slope and y-intercept of a linear equation.
- Using the x- and y-intercepts of a linear equation.

## 3.How do you graph a linear equation with two points?

Graphing linear equation with two points:

- Find the y-intercept from the linear equation and plot the point.
- From the y-intercept of the linear equation, use the slope to find the second point and plot it on the graph.
- Draw a straight line to connect the two points on the graph.

## 4.What is the formula for a linear equation?

The formula for a linear equation is \(y = mx + b\).

## 5.How do you graph a linear equation using intercepts?

To find intercepts algebraically, we use the fact that all x-intercepts have y=0 and all y-intercepts have x=0.

Determine the corresponding values of x and y by putting the values of x-intercepts and y-intercepts respectively.

## 6.What is the minimum number of points needed to graph a linear equation?

Two points are the minimum number of points needed to graph a linear equation

## 7.How do you find the Y-intercept of a graph?

The y-intercept of a graph is the point at which the graph crosses the y-axis and at y point, the x-coordinate is zero.

## 8.How do you find the intercepts of a graph?

First of all, an equation must satisfy ax+by=c. Then you just set x = 0 to find the y-intercept and set y = 0 to find the x-intercept. Then find the corresponding values.