IndietroGuided Practice: Graphing, Slope, and Equations of Lines (Beginning & Intermediate Algebra)
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Q1. Graph the equation x - y = 0 using the point-plotting method.
Background
Topic: Graphing Linear Equations in Two Variables
This question tests your ability to graph a linear equation by plotting points that satisfy the equation.
Key Terms and Formulas:
Linear Equation: An equation of the form .
Point-Plotting Method: Choose values for , solve for , and plot the resulting points.
Step-by-Step Guidance
Rewrite the equation in slope-intercept form () if possible.
Choose at least two values for (for example, and ).
For each chosen , substitute into the equation and solve for .
Plot the points you found on the coordinate plane.
Try solving on your own before revealing the answer!
Final Answer:
Rewriting gives . If , ; if , . Plot points like and , then draw a straight line through them. This is the graph of .
Q2. Graph the equation y - x = 2 using the point-plotting method.
Background
Topic: Graphing Linear Equations in Two Variables
This question tests your ability to graph a linear equation by plotting points that satisfy the equation.
Key Terms and Formulas:
Linear Equation: An equation of the form .
Point-Plotting Method: Choose values for , solve for , and plot the resulting points.
Step-by-Step Guidance
Rewrite the equation in slope-intercept form ().
Choose at least two values for (for example, and ).
For each chosen , substitute into the equation and solve for .
Plot the points you found on the coordinate plane.
Try solving on your own before revealing the answer!
Final Answer:
Rewriting gives . If , ; if , . Plot points like and , then draw a straight line through them. This is the graph of .
Q3. Graph the equation y + 2 = x using the point-plotting method.
Background
Topic: Graphing Linear Equations in Two Variables
This question tests your ability to graph a linear equation by plotting points that satisfy the equation.
Key Terms and Formulas:
Linear Equation: An equation of the form .
Point-Plotting Method: Choose values for , solve for , and plot the resulting points.
Step-by-Step Guidance
Rewrite the equation in slope-intercept form ().
Choose at least two values for (for example, and ).
For each chosen , substitute into the equation and solve for .
Plot the points you found on the coordinate plane.
Try solving on your own before revealing the answer!
Final Answer:
Rewriting gives . If , ; if , . Plot points like and , then draw a straight line through them. This is the graph of .
Q4. Graph the equation 2x - 3y = 5 using the point-plotting method.
Background
Topic: Graphing Linear Equations in Two Variables
This question tests your ability to graph a linear equation by plotting points that satisfy the equation.
Key Terms and Formulas:
Linear Equation: An equation of the form .
Point-Plotting Method: Choose values for , solve for , and plot the resulting points.
Step-by-Step Guidance
Rewrite the equation in slope-intercept form () if possible.
Choose at least two values for (for example, and ).
For each chosen , substitute into the equation and solve for .
Plot the points you found on the coordinate plane.
Try solving on your own before revealing the answer!
Final Answer:
Rewriting gives . If , ; if , . Plot points like and , then draw a straight line through them.
Q5. Graph the equation y + x = 3 using the point-plotting method.
Background
Topic: Graphing Linear Equations in Two Variables
This question tests your ability to graph a linear equation by plotting points that satisfy the equation.
Key Terms and Formulas:
Linear Equation: An equation of the form .
Point-Plotting Method: Choose values for , solve for , and plot the resulting points.
Step-by-Step Guidance
Rewrite the equation in slope-intercept form ().
Choose at least two values for (for example, and ).
For each chosen , substitute into the equation and solve for .
Plot the points you found on the coordinate plane.
Try solving on your own before revealing the answer!
Final Answer:
Rewriting gives . If , ; if , . Plot points like and , then draw a straight line through them.
Q6. Find the intercepts of the graph (equation not provided).
Background
Topic: Intercepts of Linear Equations
This question tests your ability to find the x-intercept and y-intercept of a linear equation.
Key Terms and Formulas:
x-intercept: The point where the graph crosses the x-axis ().
y-intercept: The point where the graph crosses the y-axis ().
Step-by-Step Guidance
Set in the equation and solve for to find the x-intercept.
Set in the equation and solve for to find the y-intercept.
Try solving on your own before revealing the answer!
Final Answer:
Since the equation is not provided, you would substitute to find the x-intercept and to find the y-intercept for any given equation.
Q10. Find the intercepts of the equation 2x = y.
Background
Topic: Intercepts of Linear Equations
This question tests your ability to find the x-intercept and y-intercept of a linear equation.
Key Terms and Formulas:
x-intercept: The point where the graph crosses the x-axis ().
y-intercept: The point where the graph crosses the y-axis ().
Step-by-Step Guidance
Set in and solve for to find the x-intercept.
Set in and solve for to find the y-intercept.
Try solving on your own before revealing the answer!
Final Answer:
x-intercept: Set , so . y-intercept: Set , . Both intercepts are at .
Q14. Graph the equation 2x + y = 4 by finding the intercepts.
Background
Topic: Graphing Linear Equations Using Intercepts
This question tests your ability to graph a line by finding and plotting its x- and y-intercepts.
Key Terms and Formulas:
x-intercept: Set and solve for .
y-intercept: Set and solve for .
Step-by-Step Guidance
Set in and solve for to find the x-intercept.
Set in and solve for to find the y-intercept.
Plot both intercepts on the coordinate plane.
Draw a straight line through the two points.
Try solving on your own before revealing the answer!
Final Answer:
x-intercept: , so . y-intercept: , so . Plot these points and draw the line through them.
Q19. Graph the vertical line x = 2.
Background
Topic: Graphing Vertical and Horizontal Lines
This question tests your understanding of the graphs of equations of the form or .
Key Terms and Formulas:
Vertical Line: All points where is constant.
Step-by-Step Guidance
Recognize that means every point on the line has -coordinate 2.
Plot several points with (for example, , , ).
Draw a straight vertical line through these points.
Try solving on your own before revealing the answer!
Final Answer:
The graph is a vertical line passing through on the coordinate plane.
Q20. Graph the horizontal line y = 3.
Background
Topic: Graphing Vertical and Horizontal Lines
This question tests your understanding of the graphs of equations of the form or .
Key Terms and Formulas:
Horizontal Line: All points where is constant.
Step-by-Step Guidance
Recognize that means every point on the line has -coordinate 3.
Plot several points with (for example, , , ).
Draw a straight horizontal line through these points.
Try solving on your own before revealing the answer!
Final Answer:
The graph is a horizontal line passing through on the coordinate plane.
Q5 (Section 3.3). Find the slope of the line that contains the points (3, 4) and (–2, –5).
Background
Topic: Slope of a Line Given Two Points
This question tests your ability to use the slope formula to find the slope between two points.
Key Terms and Formulas:
Slope (): The measure of steepness of a line.
Formula:
Step-by-Step Guidance
Label the points: and .
Substitute the values into the slope formula: .
Calculate the numerator: .
Calculate the denominator: .
Try solving on your own before revealing the answer!
Final Answer:
Q13 (Section 3.4). Find the equation of the line with slope 2 and y-intercept (0, –3).
Background
Topic: Slope-Intercept Form of a Line
This question tests your ability to write the equation of a line given its slope and y-intercept.
Key Terms and Formulas:
Slope-Intercept Form:
is the slope, is the y-intercept
Step-by-Step Guidance
Identify the slope () and y-intercept ().
Substitute these values into the slope-intercept form: .
Try solving on your own before revealing the answer!
Final Answer:
The equation is .
Q19 (Section 3.4). Joan makes $300 a week plus 12% commission on all sales she makes during the week. (a) Write a linear equation that calculates her weekly income, where y represents her income and x represents the amount of sales. (b) What is Joan’s weekly income if she sold $900 worth of merchandise?
Background
Topic: Writing Linear Equations from Word Problems
This question tests your ability to model a real-world situation with a linear equation and use it to make calculations.
Key Terms and Formulas:
Linear Equation:
Commission: A percentage of sales added to base pay.
Step-by-Step Guidance
Identify the fixed weekly income ($300) and the commission rate (12% or 0.12).
Let be the amount of sales. The commission is .
Write the equation: .
For part (b), substitute into your equation to find .
Try solving on your own before revealing the answer!
Final Answer:
(a) (b) . Joan’s weekly income is $408 if she sold $900 worth of merchandise.
