Table of Contents
- What do we understand by Linear equations?
- Linear equations in the standard form
- The slope of a line
- Linear equations in the slope-intercept form
- What is the x-intercept?
- What is the y-intercept?
- Graphing a Linear equation in the standard form
- Graphing a Linear equation in the slope intercept form
- Graphing a Linear equation by using the x-intercept and the y-intercept
- Solved Examples
- Frequently Asked Questions
What do we understand by Linear equations?
A linear equation is defined as an equation in which all the variables have an exponent of 1. When this equation is graphed, we can observe that it will always be in the form of a straight line.
There are two forms of representing a linear equation:
- The standard form
- The slope intercept form
Linear equations in the standard form
A linear equation with one variable is a single-variable equation. \(Ax~+~B~=~0\), is the standard form of a linear equation in one variable.
A linear equation in two variables written in the standard form is, \(Ax~+~By~=~C\).
In the equation for two variables, x and y are variables, A and B are non-zero coefficients, and C is a constant.
The slope of a line
The slope of a line indicates the steepness of a line. When a line has a positive slope, it moves upward from left to right. When moving from left to right, a line with a negative slope moves downward. Two linear functions are parallel if their slopes are the same.
The slope of a line can be calculated as the ratio of the difference between the y-coordinates, to the difference between the x-coordinates of any two points lying on the line.
The difference between the y-coordinates is called the rise, and the difference between the x-coordinates is called the run. Hence, the slope is the ratio of the rise to the run.
\(m~=~\frac{\text{Rise}}{\text{Run}}\)
The slope-intercept form of a line with a slope \(“m”\) and a y-intercept \(“b”\), or (0,b) has the equation \(y~=~mx+b\).
A horizontal line passing through \((a,~b)\) has an equation of the form \(y~=~b\). A vertical line passing through \((a,~b)\) has an equation of the form \(x~=~a\).
Linear equations in the slope-intercept form
- Consider a point of intersection where a line crosses the y-axis at \((0,~b)\).
- To find the equation of a line in the slope-intercept form, consider a random point \((x, ~y)\) on the line.
- The slope m is \(m~=~\frac{\text{change in}~y}{\text{change in}~x}~=~\frac{y-b}{x-0}\).
- When we simplify this equation further, we get \(y~=~mx+b\).
What is the x-intercept?
The x-intercept of a line is the x-coordinate of the point where the line crosses the x-axis. The y-coordinate for such a point is 0.
To find the x-intercept of an equation without drawing a graph, simply replace the value of y with zero, and solve for x. The resulting value of x is the x-intercept.
What is the y-intercept?
The y-intercept is the y-coordinate of the point where a line crosses the y-axis. The x-coordinate for such a point is 0. To find the y-intercept of an equation without drawing a graph, simply replace the value of x with zero, and solve for y. The resulting value of y is the y-intercept.
Graphing a Linear equation in the standard form
A linear equation can be graphed in the standard form using two different methods: converting the equation into the slope-intercept form and taking the help of the x and y-intercepts.
Graphing a linear equation Linear equation in the slope intercept form
One way to plot the graph of a linear equation in the standard form is to first convert it into the slope-intercept form. Let us take an example to understand this.
Plot the graph -3x + y = 1 and find the x-intercept.
Solution:
Step 1: Find the slope and the y-intercept of the given equation.
We have -3x + y = 1.
Let us express it in the slope intercept form
-3x + y =1 Write the equation
y = 1 + 3x Add 3x on each side
So the slope-intercept form is y = 3x + 1
The slope is 3 and the y-intercept is 1.
Step 2: The y-intercept is 1. So, plot the point (0,1)
Step 3: Find another point by using the slope and drawing a line.
\(m~=~\frac{\text{rise}}{\text{sun}}~=~\frac{3}{1}\)
So, we plot a point that is one unit towards the right and three units up from (0,1) which is point (1,4).
Use the two points to graph the equation.
Hence, we have the graph of the equation.
Graphing a linear equation Linear equation by using the x-intercept and the y-intercept
We can graph a linear equation by using the x-intercept and the y-intercept. Here, we have to assume y as 0 and find the coordinates for x. Similarly, assume x as 0 and find the coordinates for y. Then plot the graph using both coordinates.
Let us understand the process with an example:
Graph 6x + 2y = 4 using the x-intercept and the y-intercept form.
Solution:
Step 1: Find the x-intercept by substituting 0 for y.
6x + 2y = 4
6x + 2 (0) = 2
\(x~=~\frac{2}{6}\)
\(x~=~\frac{1}{3}\)
Find the y-intercept by substituting 0 for x.
6x + 2y = 4
6 (0) + 2y = 4
y = 2
So the points \((\frac{1}{3},~0)\) and (0,2) are plotted on the graph.
Step 2: Join the points to get the graph of the equation.
Solved Examples
Example 1: Graph -3x + 2y = 4 using the slope-intercept form.
Solution:
Step 1: Arrange the equation in the slope-intercept form
-3x + 2y = 4 Write the equation
2y = 4 + 3x Add 3x on each side
\(y~=~\frac{3}{2}x+2\) Divide each side by 2
Step 2: Find the slope and the y-intercept in the given equation
We have \(y~=~\frac{3}{2}x+2\)
Slope \(=~\frac{3}{2}\) and the y-intercept = 2
So we plot a point that is two units towards the right and three units up from (0,2) which is point (2,5)
The y-intercept is 2. So, plot (0,2)
Use the two points to graph the equation.
Hence, we have the graph of the equation.
Example 2: The equation -3x + 3y = 3 represents the distance (in miles) covered by a man in his self-driven car as well as the distance travelled by the man on his bike. Here, x represents the distance covered by the bike in miles, and y represents the distance covered by the car in miles. Draw a graph with this information. Interpret the intercepts.
Solution:
Step 1: Arrange the equation in the slope-intercept form
-3x + 3y = 3 Write the equation
3y = 3 + 3x Add 3x on each side
y = x + 1 Divide each side by 3
Step 2: Find the slope and the y-intercept in the given equation
We have y = x + 1
Slope =1 and the y-intercept =1
So we plot a point that is one unit towards the right and one unit up from (0,1) which is point (1,2)
The y-intercept is 1. So, plot (0,1).
Use the two points to graph the equation.
The x-intercept shows that the man can cover a distance of 1 mile with his bike when he is not using his car. The y-intercept shows that the distance covered by his car is 2 miles when he is not using his bike.
Example 3: Graph 2x + y = 2 using the x-intercept and the y-intercept.
Solution:
Step 1: Find the x-intercept by substituting 0 for y.
2x + y = 2
2x + 0 = 2
x = 1
Find the y-intercept by substituting 0 for x.
2x + y = 2
2 (0) + y = 2
y = 2
Step 2: Graph the equation
Example 4: Thomas has 6 dollars to buy a meal. The equation 3x + y = 6 represents the total cost of the meal, where x is the number of pizzas served by a restaurant and y is the number of burgers. Make an equation graph. Interpret the x-intercept and the y-intercept.
Solution:
Step 1: Find the x-intercept by substituting 0 for y.
3x – y = 6
3x – 0 = 6
x = 2
Find the y-intercept by substituting 0 for x.
3x + y = 6
3 (0) + y = 6
y = 6
Step 2: Graph the equation
The x-intercept shows that Thomas can buy 2 pizzas when he does not buy any burgers. The y-intercept shows that Thomas can buy 6 burgers when he does not buy any pizzas.
The point at which a line or curve crosses the axes of the graph is known as the intercept. The x-intercept is defined as a point on the line that crosses the x-axis. The y-intercept is defined as a point on the line that crosses the y-axis.
The slope of a line indicates how steep it is. The slope of a line is calculated mathematically as “rise over run”, which shows us the change in the y-coordinate divided by the change in the x-coordinate.