Podcast
Questions and Answers
Which of the following equations is in standard linear form?
Which of the following equations is in standard linear form?
- xy + z = 5
- $Ax + By = C$ (correct)
- $4x^2 + y = 9$
- $y = mx + b$
Which equation is NOT a linear equation?
Which equation is NOT a linear equation?
- x = 5
- $y = 3x - 2$
- 2x + 3y = 7
- $x^2 + y = 4$ (correct)
Which of the following equations can be rearranged into standard linear form?
Which of the following equations can be rearranged into standard linear form?
- y = 7x + 9 (correct)
- y = 4/x
- y = $\sqrt{x}$ - 2
- y = $x^3$ + 1
Which statement accurately explains why $y = x^2 + 1$ is not a linear equation?
Which statement accurately explains why $y = x^2 + 1$ is not a linear equation?
What is the slope of the line defined by the equation x = 8?
What is the slope of the line defined by the equation x = 8?
What are the values of A, B, and C in the standard form of the linear equation -5x + 2y = 3?
What are the values of A, B, and C in the standard form of the linear equation -5x + 2y = 3?
Determine which of the following equations is linear.
Determine which of the following equations is linear.
Which of the following represents a linear equation when graphed?
Which of the following represents a linear equation when graphed?
What is the x-intercept of the linear equation 4x + 3y = 12?
What is the x-intercept of the linear equation 4x + 3y = 12?
What is the y-intercept of the linear equation 2x - 5y = 10?
What is the y-intercept of the linear equation 2x - 5y = 10?
Given the equation y = -2, what is the y-intercept?
Given the equation y = -2, what is the y-intercept?
What is the x-intercept of the equation x = 4?
What is the x-intercept of the equation x = 4?
If a line is represented by the equation y = 5x + 3, where does the line intersect the y-axis?
If a line is represented by the equation y = 5x + 3, where does the line intersect the y-axis?
Given the linear equation y = 8x - 2, find the x-intercept.
Given the linear equation y = 8x - 2, find the x-intercept.
For the linear equation 6x - 3y = -18, determine the y-intercept.
For the linear equation 6x - 3y = -18, determine the y-intercept.
What is the y-intercept for the equation -3x + 5y = 9?
What is the y-intercept for the equation -3x + 5y = 9?
Given the equation -2x - 3y = 10, what is the x-intercept?
Given the equation -2x - 3y = 10, what is the x-intercept?
What is the significance of the y-intercept of a linear function in a real-world context?
What is the significance of the y-intercept of a linear function in a real-world context?
How does multiplying a linear equation by a constant affect its graph?
How does multiplying a linear equation by a constant affect its graph?
How can you determine if two linear equations have the same graph?
How can you determine if two linear equations have the same graph?
Which of the following statements is true about the graph of a linear equation with a positive slope?
Which of the following statements is true about the graph of a linear equation with a positive slope?
Which of the following is the standard form of the equation y = 5x - 3 after it has been rearranged?
Which of the following is the standard form of the equation y = 5x - 3 after it has been rearranged?
Which of the following statements about a vertical line is correct?
Which of the following statements about a vertical line is correct?
Identify which of these is needed to uniquely define any straight line on a 2D plane:
Identify which of these is needed to uniquely define any straight line on a 2D plane:
How does changing the value of 'C' in the standard form of a linear equation (Ax + By = C) affect its graph?
How does changing the value of 'C' in the standard form of a linear equation (Ax + By = C) affect its graph?
A linear equation has the form y=mx+c. If m=0, what is the result?
A linear equation has the form y=mx+c. If m=0, what is the result?
Which of the following is true about the x-intercept of a linear equation?
Which of the following is true about the x-intercept of a linear equation?
What adjustments must be made to correctly convert the equation $y = \frac{1}{3}x + 5$ into general form?
What adjustments must be made to correctly convert the equation $y = \frac{1}{3}x + 5$ into general form?
How does increasing the value of the coefficient $A$ in the standard form of a linear equation $Ax + By = C$ affect the line's steepness, assuming $B$ and $C$ remain unchanged and $A$ is positive?
How does increasing the value of the coefficient $A$ in the standard form of a linear equation $Ax + By = C$ affect the line's steepness, assuming $B$ and $C$ remain unchanged and $A$ is positive?
What characteristic do nonlinear equations possess that distinguishes them from linear equations?
What characteristic do nonlinear equations possess that distinguishes them from linear equations?
Which point represents the y-intercept of the linear equation $5x - 3y = 15$?
Which point represents the y-intercept of the linear equation $5x - 3y = 15$?
Considering a linear equation in standard form, how would the graph of $Ax + By = C$ change if $C$ were replaced with $C + k$ (where $k$ is a non-zero constant), assuming $A$ and $B$ remain unchanged?
Considering a linear equation in standard form, how would the graph of $Ax + By = C$ change if $C$ were replaced with $C + k$ (where $k$ is a non-zero constant), assuming $A$ and $B$ remain unchanged?
If the x-intercept of a linear equation is $(a, 0)$ and the y-intercept is $(0, b)$, what will the equation of the line be?
If the x-intercept of a linear equation is $(a, 0)$ and the y-intercept is $(0, b)$, what will the equation of the line be?
If a straight line has intercepts where $x$=6 and $y$=-3, what linear equation represents the same line?
If a straight line has intercepts where $x$=6 and $y$=-3, what linear equation represents the same line?
Suppose a line's equation is $y = -5$. Which of the following points does not lie on this line?
Suppose a line's equation is $y = -5$. Which of the following points does not lie on this line?
Given that a line passes through the point $(0, b)$ and $(a, 0)$, what are its x- and y-intercept respectively?
Given that a line passes through the point $(0, b)$ and $(a, 0)$, what are its x- and y-intercept respectively?
Given a linear function where if $x$ is 3, then $y$ is 5, and if $x$ is 6, then $y$ is 11, what is y when $x$ is 9?
Given a linear function where if $x$ is 3, then $y$ is 5, and if $x$ is 6, then $y$ is 11, what is y when $x$ is 9?
Flashcards
Linear Equation
Linear Equation
A linear equation is an equation of a line.
Standard form of a Linear Equation
Standard form of a Linear Equation
The standard form is Ax + By = C, where A, B, and C are constants.
X-intercept
X-intercept
The point where the graph of an equation intersects the x-axis.
Y-intercept
Y-intercept
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Finding the x-intercept
Finding the x-intercept
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Finding the y-intercept
Finding the y-intercept
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Defining the y intercept
Defining the y intercept
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Defining the x intercept
Defining the x intercept
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Study Notes
Linear Programming and Functions
- The lecture covers linear programming and linear functions.
Issues
- Timetable for lectures are Mondays and Wednesdays from 4:15pm to 5:55pm.
- Tutorials will be the following week.
- Lecture material, including the booklet, are available on the Blackboard.
- Class representatives will be chosen on Wednesday.
Content Overview
- Focus is on linear functions.
- Topics include standard form, graphing, and methods to find equations of graphed functions.
Standard Form
- A linear equation represents a line.
- The standard form of a linear equation is Ax + By = C.
Examples: Linear Equations/Functions
- 2x + 4y = 8
- 6y = 3 - x
- x = 1
- -2a + b = 5
- (4x - y) / 3 = -7
Examples: Nonlinear Equations
- 4x² + y = 5 (exponent is 2)
- √x = 4 (contains a radical)
- xy + x = 5 (variables are multiplied)
- s/r + r = 3 (variables are divided)
- These equations cannot be written in the standard form of Ax + By = C.
Determining Standard Form
- Start with equation y = 5 - 2x
- Rewrite it as 2x + y = 5
- Where A = 2, B = 1, and C = 5 to confirm it is a linear equation.
- For 2xy - 5y = 6, the term 2xy contains two variables
- The equation cannot be put in standard form and is therefore not a linear equation.
- For y = x² + 3, x is raised to the second power
- The equation cannot be written in standard form and is therefore not a linear equation.
- Start with y = 6 - 3x
- Rewrite it as 3x + y = 6
- Where A = 3, B = 1, C = 6 to confirm it is a linear equation.
- Considering (1/4)x + 5y = 3
- Multiply everything by 4 to eliminate the fraction
- Which simplifies to x + 20y = 12
- Where A = 1, B = 20, and C = 12 to confirm it is a linear equation.
- -4x+7=2 (Determine whether the equation is a linear equation, if so write it in standard form.)
X and Y Intercepts
- The x-intercept is the x-coordinate where the graph crosses the x-axis.
- The y-intercept is the y-coordinate where the graph crosses the y-axis.
Graphing Equations
- To graph 3x + 2y = 9, find the x and y intercepts.
- Set y = 0 to find the x-intercept: 3x + 2(0) = 9, so x = 3.
- Set x = 0 to find the y-intercept: 3(0) + 2y = 9, so y = 4.5.
- Plot these points, (3,0) and (0, 4.5), then connect to draw the line.
- To graph 2x + y = 4, find the x and y intercepts
- Set y = 0 to find the x-intercept: 2x + (0) = 4, so x = 2.
- Set x = 0 to find the y-intercept: 2(0) + y = 4, so y = 4.
- Plot these points, (2,0) and (0,4), then connect to draw the line.
Finding X and Y Intercepts Examples
- For x = 4y - 5, find x and y intercepts
- To find the x-intercept, set y = 0 to get x = -5 with coordinates (-5, 0).
- To find the y-intercept, set x = 0 to get y =5/4 with coordinates (0, 5/4).
- For g(x) = -3x - 1, find x and y intercepts
- To find the x-intercept, set g(x) = 0, solve for x: (-1/3, 0).
- To find the y-intercept, set x = 0, solve for g(x): (0, -1).
- For 6x - 3y = -18, find x and y intercepts
- To find the x-intercept, set y = 0 and solve for x, resulting in x = -3, giving the point (-3, 0).
- To find the y-intercept, set x = 0 and solve for y, resulting in y = 6, giving the point (0, 6).
- For x = 3, find x and y intercepts
- There is no y in the equation, the line is vertical and always equal to 3
- The x-intercept is (3, 0).
- There is no y-intercept as the line never crosses the y-axis.
- For y = -2, find x and y intercepts
- There is no x in the equation, the line is horizontal, and y is always equal to -2.
- There is no x-intercept as the line never crosses the x-axis.
- The y-intercept is (0, -2).
Example 1
- Find the x and y intercepts for y = 2x + 6
- To find the y intercept, y = 2(0) + 6, y=6.
- The y intercept coordinates are (0,6)
- To find the x intercept, 0 = 2x + 6, x = -3
- The x intercept coordinates are (-3,0)
Example 2
- Find the x and y intercepts for y = 3x + 12
- To find the y intercept, y = 3(0) + 12, y = 12
- The y intercept coordinates are (0,12)
- To find the x intercept, 0 = 3x + 12, x = -4
- The x intercept coordinates are (-4,0)
Practice Time Equations
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- y = (1/2)x + 4
-
- y = -2x + 8
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- y = -3x - 4
-
- y = 8x - 2
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- 2x + 3y = 6
-
- 5x + 2y = 10
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- -3x + 5y = 9
-
- -2x - 3y = 10
Further Practice Questions and Answers
-
- y = (1/2)x + 4
- The y intercept coordinates are (0,4) and the x intercept coordinates are (-8,0)
-
- y = -2x + 8
- The y intercept coordinates are (0,8) and the x intercept coordinates are (4,0)
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- y = -3x - 4
- The y intercept coordinates are (0,-4) and the x intercept coordinates are (-4/3,0)
-
- y = 8x - 2
- The y intercept coordinates are (0,-2) and the x intercept coordinates are (1/4,0)
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- 2x + 3y = 6
- The y intercept coordinates are (0,2) and the x intercept coordinates are (3,0)
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- 5x + 2y = 10
- The y intercept coordinates are (0,5) and the x intercept coordinates are (2,0)
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- -3x + 5y = 9
- The y intercept coordinates are (0,9/5) and the x intercept coordinates are (-3,0)
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- -2x - 3y = 10
- The y intercept coordinates are (0,-10/3) and the x intercept coordinates are (-5,0)
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