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Questions and Answers
Given the points (3, 4) and (1, 2), determine if the lines formed by these points are parallel, perpendicular, or neither.
Given the points (3, 4) and (1, 2), determine if the lines formed by these points are parallel, perpendicular, or neither.
They are perpendicular.
Write the equation of a line that is parallel to the line $y = 2x - 3$ and passes through the point (4, 1).
Write the equation of a line that is parallel to the line $y = 2x - 3$ and passes through the point (4, 1).
The equation is $y = 2x - 7$.
From the graph, determine the slope of the line that passes through the points (2, -1) and (4, 3).
From the graph, determine the slope of the line that passes through the points (2, -1) and (4, 3).
The slope is $2$.
Convert the equation $4x - 2y = 8$ into slope-intercept form.
Convert the equation $4x - 2y = 8$ into slope-intercept form.
What is the slope of a line that is perpendicular to the line defined by the equation $y = - rac{1}{3}x + 5$?
What is the slope of a line that is perpendicular to the line defined by the equation $y = - rac{1}{3}x + 5$?
If a line has a slope of $-4$ and passes through the point (2, 5), write the equation of this line in point-slope form.
If a line has a slope of $-4$ and passes through the point (2, 5), write the equation of this line in point-slope form.
Write the equation of a line that is parallel to $y = 3x + 2$ and passes through the point (-1, 4).
Write the equation of a line that is parallel to $y = 3x + 2$ and passes through the point (-1, 4).
Solve the system of equations using elimination method: $2x + 3y = 6$ and $4x - 2y = 10$. What is the value of $y$?
Solve the system of equations using elimination method: $2x + 3y = 6$ and $4x - 2y = 10$. What is the value of $y$?
In a word problem, if a snake reproduces at a rate of three babies a year, what would be the slope of the line representing the number of baby snakes over time in years?
In a word problem, if a snake reproduces at a rate of three babies a year, what would be the slope of the line representing the number of baby snakes over time in years?
Graph the inequality $y < -2x + 3$ and identify the region that represents the solution.
Graph the inequality $y < -2x + 3$ and identify the region that represents the solution.
Determine if the lines formed by the points (5, 2) and (3, 4) are parallel, perpendicular, or neither.
Determine if the lines formed by the points (5, 2) and (3, 4) are parallel, perpendicular, or neither.
If the equation of a line is given by $y = -x + 2$, write the equation of a line that is perpendicular to it and passes through the point (1, 3).
If the equation of a line is given by $y = -x + 2$, write the equation of a line that is perpendicular to it and passes through the point (1, 3).
From the graph below, identify the slope of the line segment connecting the points (0, -1) and (2, 3).
From the graph below, identify the slope of the line segment connecting the points (0, -1) and (2, 3).
Find the slope of the line represented by the equation $y = 4x - 5$.
Find the slope of the line represented by the equation $y = 4x - 5$.
Convert the standard form equation $2x + 3y = 6$ into slope-intercept form.
Convert the standard form equation $2x + 3y = 6$ into slope-intercept form.
What is the slope of a line that is parallel to the line defined by the equation $y = - rac{2}{5}x + 4$?
What is the slope of a line that is parallel to the line defined by the equation $y = - rac{2}{5}x + 4$?
Given the line $y = 3x - 1$, write the equation of a line that is parallel to it and passes through the point (2, 5).
Given the line $y = 3x - 1$, write the equation of a line that is parallel to it and passes through the point (2, 5).
If the points (6, 1) and (4, -3) define a line, what is the equation of that line in point-slope form?
If the points (6, 1) and (4, -3) define a line, what is the equation of that line in point-slope form?
In the system of equations represented by $3x + 2y = 12$ and $2x - y = 1$, use substitution to solve for x.
In the system of equations represented by $3x + 2y = 12$ and $2x - y = 1$, use substitution to solve for x.
For the word problem: A monkey climbs a tree at a rate of 5 meters per hour. What is the slope of the line representing the monkey's height over time?
For the word problem: A monkey climbs a tree at a rate of 5 meters per hour. What is the slope of the line representing the monkey's height over time?
What is the slope of a line that is parallel to the line represented by the equation $y = 2x + 1$?
What is the slope of a line that is parallel to the line represented by the equation $y = 2x + 1$?
Write the slope-intercept form of the line passing through the point (3, -2) with a slope of -5.
Write the slope-intercept form of the line passing through the point (3, -2) with a slope of -5.
Determine the slope of the line that passes through the points (-1, 4) and (3, -2).
Determine the slope of the line that passes through the points (-1, 4) and (3, -2).
If two lines intersect at a 90-degree angle, what is the relationship between their slopes?
If two lines intersect at a 90-degree angle, what is the relationship between their slopes?
Find the equation of the line in point-slope form that is perpendicular to the line $y = \frac{1}{2}x + 3$ and passes through (2, 1).
Find the equation of the line in point-slope form that is perpendicular to the line $y = \frac{1}{2}x + 3$ and passes through (2, 1).
Using the substitution method, solve the system of equations: $y = 3x + 2$ and $2x - y = 4$. What is the value of x?
Using the substitution method, solve the system of equations: $y = 3x + 2$ and $2x - y = 4$. What is the value of x?
What can you conclude if the lines represented by the equations $y = -x + 5$ and $y = -x + 1$ are graphed?
What can you conclude if the lines represented by the equations $y = -x + 5$ and $y = -x + 1$ are graphed?
In a word problem where a car travels at a constant speed, if the slope of the line representing the distance over time is 60, what does that represent?
In a word problem where a car travels at a constant speed, if the slope of the line representing the distance over time is 60, what does that represent?
Graph the inequality $y > -3x + 4$. What type of line will you draw?
Graph the inequality $y > -3x + 4$. What type of line will you draw?
What does the slope represent in the context of a line that models the growth of a population over time?
What does the slope represent in the context of a line that models the growth of a population over time?
If a line has a slope of 2, what is the slope of a line that is perpendicular to it?
If a line has a slope of 2, what is the slope of a line that is perpendicular to it?
How would you write the equation of a line that is parallel to the line $y = 4x + 3$ and passes through the point (1, 2)?
How would you write the equation of a line that is parallel to the line $y = 4x + 3$ and passes through the point (1, 2)?
What is the first step in finding the slope from the equation $y = -3x + 5$?
What is the first step in finding the slope from the equation $y = -3x + 5$?
Given the points (2, 5) and (6, 9), how do you calculate the slope of the line through them?
Given the points (2, 5) and (6, 9), how do you calculate the slope of the line through them?
In the word problem where two schools filled vans and buses with students, what system of equations could you write to represent the situation?
In the word problem where two schools filled vans and buses with students, what system of equations could you write to represent the situation?
What is the difference between graphing the system of equations using the substitution and elimination methods?
What is the difference between graphing the system of equations using the substitution and elimination methods?
Describe the process of converting the equation $3x - 5y = -10$ into slope-intercept form.
Describe the process of converting the equation $3x - 5y = -10$ into slope-intercept form.
How can you determine if two lines are parallel or perpendicular given their slopes?
How can you determine if two lines are parallel or perpendicular given their slopes?
What does it mean if you graph an inequality like $y < -2x + 4$?
What does it mean if you graph an inequality like $y < -2x + 4$?
Flashcards
Parallel Lines
Parallel Lines
Lines in a plane that never intersect.
Perpendicular Lines
Perpendicular Lines
Lines that intersect at a 90-degree angle.
Slope of a Line (from a graph)
Slope of a Line (from a graph)
Rise over run; vertical change divided by horizontal change.
Slope of a Line (from two points)
Slope of a Line (from two points)
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Slope-Intercept Form
Slope-Intercept Form
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Graphing a Line (from slope-intercept)
Graphing a Line (from slope-intercept)
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Systems of Equations (Graphing)
Systems of Equations (Graphing)
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Systems of Equations (Substitution)
Systems of Equations (Substitution)
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Parallel Line Slope
Parallel Line Slope
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Perpendicular Line Slope
Perpendicular Line Slope
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Standard to Slope-Intercept
Standard to Slope-Intercept
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Slope of Parallel Line
Slope of Parallel Line
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Slope of Perpendicular Line
Slope of Perpendicular Line
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Equation from Parallel Line & Point
Equation from Parallel Line & Point
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Equation from Perpendicular Line & Point
Equation from Perpendicular Line & Point
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Slope from Equation (Slope-Intercept)
Slope from Equation (Slope-Intercept)
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Graphing Inequalities (Slope-Intercept)
Graphing Inequalities (Slope-Intercept)
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Solving Systems by Elimination
Solving Systems by Elimination
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Solving Systems by Substitution
Solving Systems by Substitution
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Solving Systems by Graphing
Solving Systems by Graphing
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What's the slope?
What's the slope?
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Slope from Equation
Slope from Equation
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Finding the slope from two points
Finding the slope from two points
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Finding the equation from two points
Finding the equation from two points
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Slope from Graph
Slope from Graph
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Standard Form to Slope-Intercept
Standard Form to Slope-Intercept
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Slope from Two Points
Slope from Two Points
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Graphing from Slope-Intercept
Graphing from Slope-Intercept
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Study Notes
Parallel & Perpendicular Lines
- Determining Parallel/Perpendicular Lines: Parallel lines have the same slope. Perpendicular lines have slopes that are negative reciprocals of each other.
- Writing Parallel/Perpendicular Equations: Given a line, to find parallel equations maintain original slope but change y-intercept (b). For perpendicular, use the negative reciprocal of the slope and change the y-intercept (b).
- Determining Parallel/Perpendicular from Two Points: Calculate the slope of the line through the pairs of points. If slope values are the same, lines are parallel. If slopes are negative reciprocals of one another, lines are perpendicular.
Finding the Slope of a Line
- From a Graph: Count the rise over run between two points on the line. Negative rise or run indicates a negative slope.
- From Two Points: Use the formula m = (y₂ - y₁) / (x₂ - x₁). Carefully account for negative coordinates.
- From Slope-Intercept Form (y = mx + b): The slope is the coefficient 'm'.
Graphing Lines from Slope-Intercept Form
- Understanding Slope-Intercept Form: 'm' represents the slope and 'b' the y-intercept (where the line crosses the y-axis).
- Plotting: Plot the y-intercept and then use the slope to find additional points (rise/run). Start at the y-intercept and move according to the slope.
Converting Standard Form to Slope-Intercept Form
- Example: Convert 2x − 5y = 15. Solve for y.
Finding Slopes of Parallel and Perpendicular Lines
- Parallel: The slope of a line parallel to a given line has the same value as the given line's slope.
- Perpendicular: The slope of a line perpendicular to a given line is the negative reciprocal of the given line's slope.
Finding Equations of Lines
- Parallel and a Point: Find the slope of the parallel line. Use the point-slope form (y - y₁) = m(x - x₁) to write the equation, utilizing the given point.
- Perpendicular and a Point: Find the slope of the perpendicular line. Use the point-slope form to write the equation using the given point.
- Word Problems: Translate real-world scenarios into equations. Identify the initial value (y-intercept), rate of change(slope) from the information given to find the equation. Be careful with units.
- From Two Points: Use the two points to find the slope using the slope formula. Then use one of the points along with the found slope in point-slope format.
What is Slope?
- Definition: Slope measures the steepness of a line and describes how much the y-value changes for every unit of change in the x-value.
Solving Systems of Equations
- Graphing: Plot the lines and find the point of intersection.
- Substitution: Isolate one variable in one equation and substitute its expression into the other equation. Solve for one variable and substitute back.
- Elimination: Add or subtract equations to eliminate one variable. Multiply equations by constants if needed to make variable coefficients opposites.
- Word Problems: Represent the problem with two variables (x, y). Create two equations from the given information and solve using the methods of substitution or elimination.
Graphing Inequalities in Slope-Intercept Form
- Understanding Inequalities: Apply graphing procedures to inequalities as if they were equations, but consider the inequality's direction when shading the region. Use a solid line for "equal to" or "or equal to" and dashed for only "greater than" or "less than". Test a point to find the area to be shaded.
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