What are Linear Equations? [pptime time="0:01"](0:01) [/pptime]

A linear equation is an equation that represents a straight-line relationship between variables. It follows the form: $$ y = mx + c $$ where: $ m $ is the gradient (slope) , which determines the steepness of the line. $ c $ is the y-intercept , where the line crosses the y-axis. The graph of a linear equation always forms a straight line . Linear equations are fundamental in algebra and widely used in real-life applications.

What is the Gradient (m) of Linear Equations? [pptime time="0:25"](0:25) [/pptime]

$ m $ is the the gradient (also called the slope) of a linear equation determines the steepness and direction of the line. The gradient $ m $ tells us how much the value of $ y $ changes when $ x $ increases by 1. If $ m > 0 $, the line slopes upward (from left to right) If $ m < 0 $, the line slopes downward (from left to right) If $ m = 0 $, the line is horizontal The greater the absolute value of $ m $, the steeper the line. Examples: When $ m = 1 $, the line rises 1 unit up for every 1 unit right. When $ m = 3 $, the line rises 3 units up for every 1 unit right. When $ m = -2 $, the line falls 2 units down for every 1 unit right.

How to Draw a Linear Equation? [pptime time="2:05"](2:05) [/pptime]

If you know the equation, you can draw the straight line by finding two points on it. For example, given $ y = 2x + 1 $: Step 1: Find the y-intercept. When $ x = 0 $, $ y = 2 \cdot 0 + 1 = 1 $. So, the first point is $ (0,1) $. Step 2: Find another point using the gradient. Since $ m = 2 $, this means that starting from the y-intercept, moving 2 steps up and 1 step to the right, you find a second point: $ (1,3) $. Step 3: Connect both points with a straight line. Use a ruler to draw a straight line through the two points. Extend the line in both directions to represent the full equation.

Gradient and Y-Intercept in Linear Equations

🎬 Video: Understanding y = mx + c

What are Linear Equations? (0:01)

A linear equation is an equation that represents a straight-line relationship between variables. It follows the form:

$$ y = mx + c $$

where:

$ m $ is the gradient (slope), which determines the steepness of the line.
$ c $ is the y-intercept, where the line crosses the y-axis.

The graph of a linear equation always forms a straight line. Linear equations are fundamental in algebra and widely used in real-life applications.

What is the Gradient (m) of Linear Equations? (0:25)

$ m $ is the the gradient (also called the slope) of a linear equation determines the steepness and direction of the line.

The gradient $ m $ tells us how much the value of $ y $ changes when $ x $ increases by 1.

If $ m > 0 $, the line slopes upward (from left to right)
If $ m < 0 $, the line slopes downward (from left to right)
If $ m = 0 $, the line is horizontal
The greater the absolute value of $ m $, the steeper the line.

Examples:

When $ m = 1 $, the line rises 1 unit up for every 1 unit right.
When $ m = 3 $, the line rises 3 units up for every 1 unit right.
When $ m = -2 $, the line falls 2 units down for every 1 unit right.

How to Find Y-Intercept of a Linear Equation? (1:30)

The y-intercept of a linear equation is the point where the line crosses the y-axis.

The y-intercept is found by setting $ x = 0 $ in the equation:

$$ y = m \cdot 0 + c = c $$

This means the y-intercept at the point $ (0, c) $. The y-intercept determines where the line starts on the y-axis, shifting the line up or down without changing its slope.

How to Draw a Linear Equation? (2:05)

If you know the equation, you can draw the straight line by finding two points on it.

For example, given $ y = 2x + 1 $:

Step 1: Find the y-intercept.
When $ x = 0 $, $ y = 2 \cdot 0 + 1 = 1 $. So, the first point is $ (0,1) $.
Step 2: Find another point using the gradient.
Since $ m = 2 $, this means that starting from the y-intercept, moving 2 steps up and 1 step to the right, you find a second point: $ (1,3) $.
Step 3: Connect both points with a straight line.
Use a ruler to draw a straight line through the two points. Extend the line in both directions to represent the full equation.