Linear relationships
Mathematics, Grade 8
Linear relationships
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Linear relationships
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Linear relationships
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Linear relationships
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Linear relationships
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Study Guide Linear relationships Mathematics, Grade 8
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LINEAR RELATIONSHIPS Linear relationships refer to two quantities that are related with a linear equation. Since a linear equation is a line, a linear relationship refers to two quantities on a line and their relationship to one another. This relationship can be direct or inverse. • Direct variation refers to two variables that are related because the ratio of their values is always the same. If y varies directly as x, it means if y is doubled, then x is doubled. The formula for a direct variation is y = kx, where k is the constant of variation. • Linear relationships can also be represented when graphed. The graph of a linear inequality can also be graphed. When a linear inequality is graphed, the solution set contains the points in the shaded area of the graph. • Scatter plots also show the relationship between two variables. A line of best fit shows the pattern of the points on a scatter plot. From the line of best fit, conclusions can be made about the scatter plot. How to use linear relationships Linear relationships refer to two quantities that are related with a linear equation. When two variables are in direct variation, the ratio of their values remains the same. The table shows a direct variation. What is the constant of variation? © Copyright NewPath Learning. All Rights Reserved. Permission is granted for the purchaser to print copies for non-commercial educational purposes only. Visit us at www.NewPathLearning.com.
The table shows a direct variation, which means that y varies directly as x. The ratio of their values are 1/2 = 2/4 = 3/6 = 4/8 = 5/10, the constant of variation is 1/2. If it is known that y varies directly as x, and values are given, such as in the table, other ratios could be found. For example, using the table, what is y when x is 38? Ex. Since y varies directly as x, 1/2 = y/38 → 38 = 2y → y = 19 This linear relationship could be graphed by using the correct x and y values. Linear inequalities can also be graphed. Linear inequalities are graphed by replacing the inequality sign with an equal sign and graphing the line. When the line is graphed, it will either be solid if ≤ or ≥, or dashed if < or >. To represent the solution set, the area either above the line is shaded if > or ≥, or the area below the line is shaded if < or ≤. A point in the solution set should be check to make sure the inequality is correct. For example, the inequality shown is y ≤ x – 4. Is it graphed correctly? The inequality is graphed correctly because it represents the line y = x - 4, has a solid line for the ≤, and is shaded below the line. When the point (4, -2) is checked, the inequality is true. Remember to make sure the line is in the form of y = mx + b before graphing in order to get the correct answer. © Copyright NewPath Learning. All Rights Reserved. Permission is granted for the purchaser to print copies for non-commercial educational purposes only. Visit us at www.NewPathLearning.com.
Try This! 1. If y varies directly as x and y = 12 when x = 8, what is y when x = 24? 2. If y varies directly as x and y = 18 when x = 6, what is the constant of variation? 3. If the inequality, y > 2x + 5, is to be graphed, is the line dashed or solid? 4. If the inequality, -3y ≤ x + 6, is to be graphed, is the shaded area above or below the line? 5. Based on the scatter plot, if a line of best fit were drawn, how much would a candy bar cost in the 2001? © Copyright NewPath Learning. All Rights Reserved. Permission is granted for the purchaser to print copies for non-commercial educational purposes only. Visit us at www.NewPathLearning.com.
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