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Questions and Answers
For the rule $y = x + 2$, the basic shape is __.
For the rule $y = x + 2$, the basic shape is __.
linear
For the rule $y = 2x - 2$, the gradient is __.
For the rule $y = 2x - 2$, the gradient is __.
2
For the rule $y = -2x + 1$, the y-intercept is __.
For the rule $y = -2x + 1$, the y-intercept is __.
1
What is the gradient of the line $y = 5x - 2$?
What is the gradient of the line $y = 5x - 2$?
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What is the y-intercept of the line $y = -4x$?
What is the y-intercept of the line $y = -4x$?
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What is the equation of a line from the table of values?
What is the equation of a line from the table of values?
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Does the point (0, 4) lie on the line $y = 2x - 4$?
Does the point (0, 4) lie on the line $y = 2x - 4$?
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What is the gradient of the line $-6x + 2y = 10$?
What is the gradient of the line $-6x + 2y = 10$?
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How long does it take Hamish to walk home from school?
How long does it take Hamish to walk home from school?
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What is the gradient of the line drawn in part b of Hamish's walking?
What is the gradient of the line drawn in part b of Hamish's walking?
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If Hamish walks for 90 seconds, how far is he from home?
If Hamish walks for 90 seconds, how far is he from home?
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How long has Hamish been walking if he is 315 m away from home?
How long has Hamish been walking if he is 315 m away from home?
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Study Notes
Linear Equations in the Form y = mx + c
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y = mx + c is the standard form of a linear equation where:
- m represents the gradient (slope) of the line
- c represents the y-intercept (where the line crosses the y-axis)
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Gradient (m):
- In the equation y = mx + c, the coefficient of x represents the gradient.
- A positive gradient indicates an upward sloping line.
- A negative gradient indicates a downward sloping line.
- Zero gradient means the line is horizontal.
-
Y-intercept (c):
- In the equation y = mx + c, the constant term represents the y-intercept.
- This is the point where the line crosses the y-axis.
Finding the Equation of a Line
- Given two points on a line, calculate the gradient (m) using the formula: m = (y2 - y1) / (x2 - x1).
- Find the y-intercept (c) by substituting one of the points and the calculated gradient into the equation y = mx + c, and solve for c.
Plotting Points and Graphing Lines
- To plot a point on a coordinate plane, use its coordinates (x, y):
- The first coordinate (x) represents the horizontal position.
- The second coordinate (y) represents the vertical position.
- To graph a linear equation:
- Find at least two points that satisfy the equation.
- Plot these points on the coordinate plane.
- Draw a straight line passing through these points.
Horizontal and Vertical Lines
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Horizontal lines:
- Have equations in the form y = c (constant).
- Their gradient is 0.
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Vertical lines:
- Have equations in the form x = c (constant).
- Their gradient is undefined.
Finding the Gradient from a Graph
- Gradient is the measure of the steepness of a line.
- It can be calculated by dividing the change in y (vertical change) by the change in x (horizontal change) between any two points on the line.
Determining if a Point Lies on a Line
- Substitute the coordinates of the point into the equation of the line.
- If the equation is true, the point lies on the line.
Rearranging Linear Equations:
- The goal is to isolate the y-term on one side of the equation and have the equation in the form y = mx + c.
- Use basic algebraic operations to rearrange the equation:
- Add or subtract terms to move them to the other side of the equation.
- Multiply or divide both sides of the equation by the same non-zero number to isolate the y-term.
X-Intercept
- The x-intercept is the point where the line crosses the x-axis.
- At the x-intercept, the y-coordinate is 0.
- To find the x-intercept set y = 0 in the equation and solve for x.
Y-Intercept
- The y-intercept is the point where the line crosses the y-axis.
- At the y-intercept, the x-coordinate is 0.
- To find the y-intercept, set x = 0 in the equation and solve for y.
Word Problems
- Understand the context of the problem and translate the information into mathematical relationships.
- Identify the independent and dependent variables.
- Use the given information to create a table of values or a graph.
- Determine the equation of the line representing the relationship between the variables.
- Use the equation to solve the given questions.
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Description
This quiz covers the fundamentals of linear equations in the form y = mx + c, focusing on understanding gradients, y-intercepts, and how to find the equation of a line from given points. Test your knowledge on plotting points and graphing lines as well. Perfect for students studying algebra.
