Geometry: Parallel and Perpendicular Lines

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Questions and Answers

Given two points A(2, 3) and B(4, 7), determine whether the lines formed by these points are parallel, perpendicular, or neither.

The lines are neither parallel nor perpendicular.

Write the equation of a line that is parallel to the line given by $y = 3x + 1$ and passes through the point (1, -2).

The equation is $y = 3x - 5$.

From the standard form equation $2x - 3y = 6$, convert it to slope-intercept form.

The slope-intercept form is $y = \frac{2}{3}x - 2$.

In the context of a real-world problem, if a line representing the total cost C of renting vehicles can be modeled as $C = 150v + 200b$, where v is the number of vans and b is the number of buses, what is the slope of this equation?

<p>The slope is 150.</p> Signup and view all the answers

Solve the system of equations using elimination: 2x + 3y = 12 and 4x - 3y = 6.

<p>The solution is (3, 0).</p> Signup and view all the answers

How do you determine if two lines given by coordinates are perpendicular?

<p>If the product of their slopes is -1, then the lines are perpendicular.</p> Signup and view all the answers

What is the slope of a line parallel to the line represented by the equation $y = -2x + 5$?

<p>-2</p> Signup and view all the answers

When given two points, how can you find the slope of the line that connects them?

<p>Use the formula $m = \frac{y_2 - y_1}{x_2 - x_1}$ where $(x_1, y_1)$ and $(x_2, y_2)$ are the points.</p> Signup and view all the answers

In the context of a word problem, if you have a total of 48 apples and double the amount collected in a second basket can be represented as $y = 2x$, what is the slope?

<p>2</p> Signup and view all the answers

What steps do you take to graph the inequality $y < 3x - 2$ in slope-intercept form?

<p>First, graph the line $y = 3x - 2$ as a dashed line, then shade below the line.</p> Signup and view all the answers

When given the equation of a line in slope-intercept form, what do you understand the slope to indicate about the line?

<p>The slope indicates the rate of change of the y-value for every unit change in the x-value, showing the steepness and direction of the line.</p> Signup and view all the answers

What must be true about the slopes of two lines if they are parallel?

<p>The slopes of two parallel lines must be equal.</p> Signup and view all the answers

Given two points, how do you calculate the slope of the line that connects them?

<p>The slope can be calculated using the formula $m = \frac{y_2 - y_1}{x_2 - x_1}$, where $(x_1, y_1)$ and $(x_2, y_2)$ are the two points.</p> Signup and view all the answers

In a word problem involving vehicles, how can you derive the equations representing the situation?

<p>You can use the number of vehicles and the total related to the cost or capacity to set up equations based on the relationships described.</p> Signup and view all the answers

How do you convert an equation from standard form to slope-intercept form?

<p>To convert to slope-intercept form, solve for y to isolate it, resulting in the format $y = mx + b$, where m is the slope.</p> Signup and view all the answers

What is the significance of the slope in a linear equation, and how can it impact the relationship between two variables?

<p>The slope represents the rate of change between the two variables, indicating how much one variable changes for a unit change in the other.</p> Signup and view all the answers

When presented with a system of equations, what methods can be used to find their intersection points, and how do you determine which method is most effective?

<p>You can use graphing, substitution, or elimination. The choice depends on the complexity of the equations and personal preference for visual versus algebraic solutions.</p> Signup and view all the answers

Describe how to determine if two lines are parallel or perpendicular given their equations in slope-intercept form.

<p>Lines are parallel if they have the same slope and perpendicular if their slopes are negative reciprocals of each other.</p> Signup and view all the answers

Explain the process of converting an equation from standard form to slope-intercept form, illustrating with an example.

<p>To convert, isolate y; for example, from $2x - 3y = 6$, rearranging gives $y = rac{2}{3}x - 2$.</p> Signup and view all the answers

In a word problem involving vehicles, how can you derive the slope and the equation of the line that models the relationship between the number of vehicles and capacity?

<p>Set up an equation based on total capacity in relation to the number of vehicles; for instance, $C = mv + nb$, where m is capacity per van and n is capacity per bus.</p> Signup and view all the answers

If two lines have slopes of 2 and -0.5, are they parallel, perpendicular, or neither?

<p>They are perpendicular because the product of their slopes is -1.</p> Signup and view all the answers

Explain how to find the slope of a line from its equation in slope-intercept form, such as $y = 4x - 3$.

<p>The slope is the coefficient of x, which is 4.</p> Signup and view all the answers

In a word problem about renting vehicles, if a class of students is represented by the equation $y = 5x + 10$, what does the slope indicate?

<p>The slope of 5 indicates that for each additional vehicle rented, there are 5 more students.</p> Signup and view all the answers

How can you determine the equation of a line that is perpendicular to the line with the equation $y = 2x + 1$ and passes through the point (3, -2)?

<p>The slope of the perpendicular line is -0.5, so the equation is $y + 2 = -0.5(x - 3)$, which simplifies to $y = -0.5x - 0.5$.</p> Signup and view all the answers

If you convert the standard form $3x + 4y = 12$ to slope-intercept form, what is the slope?

<p>The slope is -0.75.</p> Signup and view all the answers

What is the slope of the line represented by the equation $y = -5x + 7$?

<p>-5 (C)</p> Signup and view all the answers

Which of the following pairs of lines are perpendicular?

<p>2 (A)</p> Signup and view all the answers

What is the equation of the line that is parallel to the line $y = 2x - 4$ and passes through the point (2, 3)?

<p>y = 2x + 1 (B)</p> Signup and view all the answers

From two points A(-1, 2) and B(3, -2), what is the slope of the line connecting them?

<p>-2 (A)</p> Signup and view all the answers

If the equation of a line in standard form is $3x + 2y = 12$, what is the slope when it is converted to slope-intercept form?

<p>-1.5 (C)</p> Signup and view all the answers

How would you find the slope of a line from the graph of the line?

<p>You can find the slope by choosing two points on the line, calculating the change in y-values over the change in x-values, or using the formula $m = \frac{y_2 - y_1}{x_2 - x_1}$.</p> Signup and view all the answers

If you are given two points (3, 4) and (7, 8), how do you calculate the slope of the line through these points?

<p>Use the formula $m = \frac{8 - 4}{7 - 3} = 1$.</p> Signup and view all the answers

What is the slope of a line that is perpendicular to a line with a slope of 3?

<p>The slope of the perpendicular line would be $-\frac{1}{3}$, since perpendicular slopes are negative reciprocals.</p> Signup and view all the answers

Explain how to find the equation of a line given a point and a line that is parallel to it.

<p>First, identify the slope of the parallel line; then use the point-slope form $y - y_1 = m(x - x_1)$ to write the equation.</p> Signup and view all the answers

Define slope in the context of a linear equation.

<p>Slope is the ratio of the vertical change to the horizontal change between any two points on the line, often denoted as $m$.</p> Signup and view all the answers

Describe the process of converting an equation from standard form to slope-intercept form.

<p>Rearrange the equation to isolate $y$ on one side, resulting in the form $y = mx + b$, where $m$ is the slope and $b$ is the y-intercept.</p> Signup and view all the answers

How would you identify whether two lines are parallel from their equations?

<p>Two lines are parallel if they have the same slope; check the coefficients of $x$ in their equations after converting to slope-intercept form.</p> Signup and view all the answers

When given the equation $y = -2x + 5$, what is the slope and what does it indicate?

<p>The slope is $-2$, indicating that the line descends at a rate of 2 units down for every unit it moves to the right.</p> Signup and view all the answers

How can you write the equation of a line from a word problem about a measurable relationship?

<p>Identify the variables and their rates of change, then translate this into a linear equation, often in the form $y = mx + b$.</p> Signup and view all the answers

In a scenario where a line passes through (4, -1) and is perpendicular to a line with slope 1, what is the equation of the new line?

<p>First, find the slope of the new line, which is -1, then use point-slope form: $y + 1 = -1(x - 4)$.</p> Signup and view all the answers

Flashcards

Parallel lines

Lines that never intersect and have the same slope.

Slope-intercept form

A way to write a linear equation as y = mx + b, where m is the slope and b is the y-intercept.

Solving systems of equations by graphing

Finding the point where two or more lines intersect on a graph.

Slope

The steepness of a line, calculated by rise over run ((y2 - y1) / (x2 - x1)).

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Perpendicular lines

Lines that intersect at a 90-degree angle. Their slopes are negative reciprocals of each other.

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Finding the slope from two points

The steepness of a line calculated by the change in y divided by the change in x: (y2 - y1) / (x2 - x1).

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Solving systems of equations by substitution

Solving for one variable in terms of the other and substituting that expression into the second equation to solve for the other variable.

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Slope from Two Points

The steepness of a line calculated by the change in y divided by the change in x: (y2 - y1) / (x2 - x1).

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Equation from Two Points

Use the two points to find the slope, then substitute one point and the slope into the point-slope form (y - y1) = m(x - x1) to get the equation.

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Graphing Inequalities

Plot the line as if it were an equality. Then, use a test point to determine which side of the line satisfies the inequality. Shade the solution area.

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Solving Systems by Elimination

Multiply equations by constants to make the coefficients of one variable opposites. Add the equations together to eliminate one variable. Solve for the remaining variable and substitute back to solve for the other.

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Writing Equation from Two Points

Use the two points to determine the slope. Then use one point and the slope in the point-slope form (y - y1) = m(x - x1) to find the equation.

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Slope of Perpendicular Lines

The slopes of perpendicular lines are negative reciprocals of each other. This means if one line has a slope of 'm', the perpendicular line has a slope of '-1/m'.

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Equation from Parallel Line & Point

To find the equation of a line parallel to a given line and passing through a point, use the same slope as the given line and substitute the point into the point-slope form (y - y1) = m(x - x1).

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Equation from Word Problem

Translate the word problem into a system of equations representing the relationships given. Solve the system to find the unknowns.

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Solving Systems by Graphing

Finding the point where two or more lines intersect on a graph.

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Solving Systems by Substitution

Solving for one variable in terms of the other and substituting that expression into the second equation to solve for the other variable.

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Equation from Perpendicular Line & Point

To find the equation of a line perpendicular to a given line and passing through a point, use the negative reciprocal of the slope of the given line and substitute the point into the point-slope form (y - y1) = m(x - x1).

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Study Notes

Parallel and Perpendicular Lines

  • Determining Parallel/Perpendicular Lines: Parallel lines have equal slopes; perpendicular lines have slopes that are negative reciprocals of each other. If the slopes are neither equal nor negative reciprocals, the lines are neither parallel nor perpendicular.
  • Writing Parallel/Perpendicular Equations: To write a parallel equation, use the given slope and a point. To write a perpendicular equation, use the negative reciprocal of the given slope and a point.
  • Determining Parallel/Perpendicular Lines from Points: Calculate the slope between the two points given for each line. Compare the slopes to determine if the lines are parallel, perpendicular, or neither. If two lines have the same slope, they are parallel. If the product of their slopes is -1, they are perpendicular.

Finding the Slope of a Line

  • From a Graph: Count the rise and run between two points on the line. Rise over run equals slope.
  • From Two Points: Use the formula: (y₂ - y₁) / (x₂ - x₁)
  • From an Equation (Slope-Intercept Form): The slope-intercept form is y = mx + b, where 'm' is the slope.
  • Graphing Lines from Slope-Intercept Form: Use the y-intercept as a starting point, then use the slope to find other points.

Converting Standard Form to Slope-Intercept Form

  • To write an equation in standard form (Ax + By = C) into slope-intercept form (y = mx + b) solve for "y."

Finding Slopes of Parallel/Perpendicular Lines

  • Parallel Lines: The slope of a line parallel to a given line is equal to the slope of the given line.
  • Perpendicular Lines: The slope of a line perpendicular to a given line is the negative reciprocal of the slope of the given line.

Finding Equations of Lines

  • From Parallel Line and a Point: Use the slope of the parallel line and the given point to find the equation.
  • From Perpendicular Line and a Point: Use the negative reciprocal of the perpendicular line's slope and the given point to find the equation.
  • Word Problems: Identify the initial value (y-intercept) and the rate of change (slope). Form an equation based on the information given in the word problem, then solve for the variables. For example, for a growth rate problem, the slope represents the growth rate and is multiplied by the time.

What is Slope?

  • Slope represents the steepness and direction of a line. A positive slope indicates an upward trend, a negative slope a downward trend, and a zero slope indicates a horizontal line.

Systems of Equations

  • Graphing: Graph both equations and find the point where they intersect.
  • Substitution: Solve one equation for one variable, then substitute that expression into the other equation and solve.
  • Elimination: Multiply one or both equations by constants to eliminate a variable when adding or subtracting the equations.
  • Word Problems: Define variables to represent unknown quantities. Form equations based on the relationships described in the problem to solve for the unknown values, then solve the system of equations using any method.

Graphing Inequalities in Slope-Intercept Form

  • Graph the corresponding equation using the slope and y-intercept, then determine whether to shade above or below the line based on the inequality sign (greater than or less than). A dashed line indicates the inequality does not contain equality. A solid line indicates the inequality includes equality (greater than or equal to, less than or equal to).

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