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Linear Graph: Definition and Examples | EDU.COMEDU.COMResourcesBlogGuidePodcastPlanBackHomesvg]:size-3.5">Math Glossarysvg]:size-3.5">Linear GraphLinear Graph: Definition and ExamplesTable of ContentsLinear Graph Definition of Linear Graph

A linear graph is a graphical representation that shows the relationship between two or more quantities using straight lines. The term 'linear' refers to straight, which means these graphs form straight lines to display relationships between different quantities. Linear graphs help in showing results in single straight lines without using curves, dots, or bars.

A linear graph is represented by the equation y=mx+cy = mx + cy=mx+c, where mmm is the gradient (slope) of the graph and ccc is the y-intercept (where the line crosses the y-axis). This equation can also be written as ax+by+c=0ax + by + c = 0ax+by+c=0, where aaa, bbb, and ccc are constants. A linear equation has two variables with many solutions and can extend to an infinite number of points on the line. All points on a linear graph are collinear, meaning they all lie on the same straight line.

Examples of Linear Graph Example 1: Identifying Slope and Y-intercept from Linear Equations Problem:

For the linear equation y=4x−7y = 4x - 7y=4x−7, identify the slope and y-intercept, and explain what they represent on the graph.

Step-by-step solution:

Step 1, Recall that the standard form of a linear equation is y=mx+cy = mx + cy=mx+c, where mmm is the slope and ccc is the y-intercept.

Step 2, Compare the given equation y=4x−7y = 4x - 7y=4x−7 with the standard form:

y=4x−7y = 4x - 7y=4x−7 y=mx+cy = mx + cy=mx+c

Step 3, Identify the slope (mmm) by looking at the coefficient of xxx:

m=4m = 4m=4

Step 4, Identify the y-intercept (ccc) by looking at the constant term:

c=−7c = -7c=−7

Step 5, Explain what these values represent:

The slope of 444 means that for every 111 unit increase in xxx, yyy increases by 444 units The y-intercept of −7-7−7 means the line crosses the y-axis at the point (000, −7-7−7) Example 2: Converting General Form to Slope-Intercept Form Problem:

Convert the linear equation 3x+2y−6=03x + 2y - 6 = 03x+2y−6=0 to slope-intercept form (y=mx+cy = mx + cy=mx+c) and identify the slope and y-intercept.

Step-by-step solution:

Step 1, Start with the given equation in general form:

3x+2y−6=03x + 2y - 6 = 03x+2y−6=0

Step 2, Isolate the yyy term by subtracting 3x3x3x from both sides:

3x+2y−6−3x=0−3x3x + 2y - 6 - 3x = 0 - 3x3x+2y−6−3x=0−3x 2y−6=−3x2y - 6 = -3x2y−6=−3x

Step 3, Add 666 to both sides:

2y−6+6=−3x+62y - 6 + 6 = -3x + 62y−6+6=−3x+6 2y=−3x+62y = -3x + 62y=−3x+6

Step 4, Divide both sides by 222 to solve for yyy:

2y2=−3x+62\frac{2y}{2} = \frac{-3x + 6}{2}22y​=2−3x+6​ y=−3x+62y = \frac{-3x + 6}{2}y=2−3x+6​ y=−32x+3y = -\frac{3}{2}x + 3y=−23​x+3

Step 5, Identify the slope and y-intercept from the slope-intercept form:

The slope m=−32m = -\frac{3}{2}m=−23​, which means for every 222 units increase in xxx, yyy decreases by 333 units The y-intercept c=3c = 3c=3, which means the line crosses the y-axis at the point (000, 333) Example 3: Solving for y Using a Linear Equation Problem:

Substitute −2-2−2 for xxx and find the result for yyy in the equation y=3x+1y = 3x + 1y=3x+1.

Step-by-step solution: Step 1, Start with the given linear equation: y=3x+1y = 3x + 1y=3x+1 Step 2, Replace xxx with the given value (−2-2−2): y=3(−2)+1y = 3(-2) + 1y=3(−2)+1 Step 3, Multiply 333 by −2-2−2: y=−6+1y = -6 + 1y=−6+1 Step 4, Add −6-6−6 and 111: y=−5y = -5y=−5 Comments(3)SScienceTutorCodyNovember 6, 2025I've used this clear linear graph definition to help my students. It made understanding slopes and y - intercepts a breeze!

JJewelryDesignerZachNovember 4, 2025This clear def of linear graph helped my students grasp the concept quickly. It's a great resource for teaching math!

MMsAdventurerSeptember 17, 2025I’ve been using this page to help my kids understand linear graphs, and the clear definition and examples made it so much easier for them to grasp. Great resource!

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