BackBeginning & Intermediate Algebra: Coordinate Graphing, Slope, and Linear Equations Study Guide
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Q1. Plot the given point in a rectangular coordinate system: (6, -2)
Background
Topic: Plotting Points in the Coordinate Plane
This question tests your ability to locate and plot a point given its coordinates (x, y) on a Cartesian plane.
Key Terms and Concepts:
Coordinate Plane: A two-dimensional surface formed by two perpendicular number lines (x-axis and y-axis).
Ordered Pair (x, y): The first number is the x-coordinate (horizontal), and the second is the y-coordinate (vertical).
Step-by-Step Guidance
Identify the x-coordinate (6) and y-coordinate (-2) from the ordered pair.
Start at the origin (0, 0) on the graph.
Move 6 units to the right along the x-axis (since x is positive).
From that point, move 2 units down (since y is negative).
Mark the point where you end up. This is the location of (6, -2).
Try solving on your own before revealing the answer!
Final Answer: The correct graph is option D.
Option D correctly plots the point (6, -2) by moving 6 units right and 2 units down from the origin.
Q2. Give the ordered pairs that correspond to the points labeled in the figure.
Background
Topic: Reading Coordinates from a Graph
This question checks your ability to read and write the coordinates of points shown on a Cartesian plane.
Key Terms and Concepts:
Ordered Pair: The coordinates (x, y) that describe the position of a point on the graph.
Step-by-Step Guidance
Look at each labeled point on the graph.
For each point, count how many units you move horizontally from the origin to reach the point (this is x).
Then, count how many units you move vertically from the origin to reach the point (this is y).
Write the coordinates as (x, y) for each labeled point.
Try solving on your own before revealing the answer!
Final Answer: The correct answer is A: C(-2, 4), D(3, -2), E(0, -3), F(-4, 0).
Each point's coordinates are determined by counting units from the origin along the x and y axes.
Q3. Determine whether the ordered pair is a solution of the given equation: 3x + 4y = 12; (4, 0)
Background
Topic: Solutions to Linear Equations
This question tests your ability to check if a given ordered pair satisfies a linear equation.
Key Terms and Formulas:
Solution to an Equation: An ordered pair (x, y) that makes the equation true when substituted.
Step-by-Step Guidance
Substitute x = 4 and y = 0 into the equation 3x + 4y = 12.
Calculate the left side: 3(4) + 4(0).
Simplify the expression to see if it equals 12.
Try solving on your own before revealing the answer!
Final Answer: No, (4, 0) is not a solution.
Substituting gives 3(4) + 4(0) = 12, which simplifies to 12 = 12, so actually, (4, 0) is a solution. (Correction: Yes, (4, 0) is a solution.)
Q4. Find a solution to the equation using the value given for x: y = 2x + 1; x = 3
Background
Topic: Evaluating Linear Equations
This question checks your ability to substitute a value for x into a linear equation to find the corresponding y value.
Key Terms and Formulas:
Substitution: Replacing a variable with a given value.
Step-by-Step Guidance
Write the equation: y = 2x + 1.
Substitute x = 3 into the equation.
Calculate y by performing the arithmetic.
Try solving on your own before revealing the answer!
Final Answer: y = 7, so the solution is (3, 7).
Substituting x = 3 gives y = 2(3) + 1 = 6 + 1 = 7.
Q5. Use the graph to identify the x- and y-intercepts or state that there is no x- or y-intercept.
Background
Topic: Intercepts of a Line
This question tests your ability to read a graph and identify where the line crosses the x-axis and y-axis.
Key Terms and Concepts:
x-intercept: The point where the line crosses the x-axis (y = 0).
y-intercept: The point where the line crosses the y-axis (x = 0).
Step-by-Step Guidance
Look at the graph and find the point where the line crosses the x-axis. Record the x-coordinate.
Find the point where the line crosses the y-axis. Record the y-coordinate.
Write the intercepts as (x, 0) for the x-intercept and (0, y) for the y-intercept.
Try solving on your own before revealing the answer!
Final Answer: x-intercept = 4, y-intercept = 8
The line crosses the x-axis at (4, 0) and the y-axis at (0, 8).
Q6. Find the x-intercept and y-intercept of the graph of the equation. Do not graph the equation: 3x + 2y = 6
Background
Topic: Finding Intercepts Algebraically
This question tests your ability to find intercepts by setting variables to zero in a linear equation.
Key Terms and Formulas:
x-intercept: Set y = 0 and solve for x.
y-intercept: Set x = 0 and solve for y.
Step-by-Step Guidance
To find the x-intercept, set y = 0 in the equation and solve for x.
To find the y-intercept, set x = 0 in the equation and solve for y.
Write the intercepts as ordered pairs.
Try solving on your own before revealing the answer!
Final Answer: x-intercept = (2, 0), y-intercept = (0, 3)
Setting y = 0 gives 3x = 6, so x = 2. Setting x = 0 gives 2y = 6, so y = 3.
Q7. Write an equation for the graph.
Background
Topic: Writing Equations of Lines
This question tests your ability to write the equation of a line from its graph, usually in slope-intercept form (y = mx + b).
Key Terms and Formulas:
Slope-intercept form: where m is the slope and b is the y-intercept.
Step-by-Step Guidance
Identify the y-intercept (where the line crosses the y-axis).
Find the slope by choosing two points on the line and calculating the rise over run.
Write the equation using the slope and y-intercept.
Try solving on your own before revealing the answer!
Final Answer: y = 2
The line is horizontal and crosses the y-axis at 2, so the equation is y = 2.
Q8. Graph the equation: 2y = 5x + 8
Background
Topic: Graphing Linear Equations
This question tests your ability to graph a linear equation by finding intercepts or rewriting in slope-intercept form.
Key Terms and Formulas:
Slope-intercept form:
Step-by-Step Guidance
Solve for y to put the equation in slope-intercept form.
Identify the slope (m) and y-intercept (b).
Plot the y-intercept on the graph.
Use the slope to find another point on the line.
Try solving on your own before revealing the answer!
Final Answer: Option C is the correct graph.
After solving for y, the equation is y = (5/2)x + 4. The correct graph matches this slope and y-intercept.
Q9. Find the slope of the line passing through the pair of points or state that the slope is undefined: (2, 4) and (2, -1)
Background
Topic: Slope of a Line
This question tests your ability to calculate the slope between two points and recognize when the slope is undefined.
Key Terms and Formulas:
Slope formula:
Step-by-Step Guidance
Label the points as (x1, y1) and (x2, y2).
Substitute the values into the slope formula.
Check if the denominator is zero (which would make the slope undefined).
Try solving on your own before revealing the answer!
Final Answer: The slope is undefined.
Both points have the same x-value, so the line is vertical and the slope is undefined.
Q10. Put the equation in slope-intercept form by solving for y. Use the slope and y-intercept to graph the equation: 2x + 3y = 12
Background
Topic: Slope-Intercept Form and Graphing
This question tests your ability to rearrange a linear equation into y = mx + b form and use it to graph the line.
Key Terms and Formulas:
Slope-intercept form:
Step-by-Step Guidance
Start with the equation 2x + 3y = 12.
Solve for y by isolating it on one side of the equation.
Identify the slope (m) and y-intercept (b) from the resulting equation.
Plot the y-intercept on the graph and use the slope to find another point.
Try solving on your own before revealing the answer!
Final Answer: Option B is the correct graph.
Solving for y gives y = -2/3x + 4. The correct graph matches this slope and y-intercept.
Q11. Find the point-slope form of the equation of the line satisfying the given conditions and use this to write the slope-intercept form of the equation: Slope = 2, passing through (3, -5)
Background
Topic: Point-Slope and Slope-Intercept Forms
This question tests your ability to write the equation of a line given a point and a slope, and then convert it to slope-intercept form.
Key Terms and Formulas:
Point-slope form:
Slope-intercept form:
Step-by-Step Guidance
Write the point-slope form using m = 2 and (x1, y1) = (3, -5).
Expand and simplify the equation to get it into slope-intercept form.
Identify the slope and y-intercept in the final equation.
Try solving on your own before revealing the answer!
Final Answer: y = 2x - 11
Using point-slope form and simplifying gives the slope-intercept form y = 2x - 11.
