College Algebra Review: Linear Functions, Formulas, and Systems

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Q1. Given the two points (-12, -13) and (18, 27), find the following: Midpoint, Distance, Slope, Y-Intercept, Equation, Graph

Background

Topic: Linear Equations and Coordinate Geometry

This question tests your understanding of how to work with points on the coordinate plane, including finding the midpoint, distance, slope, y-intercept, and the equation of the line passing through two points.

Key Terms and Formulas

Midpoint Formula:
Distance Formula:
Slope Formula:
Point-Slope Form:
Slope-Intercept Form:

Step-by-Step Guidance

Label your points: Let and .
Find the midpoint using the midpoint formula:
Calculate the distance between the points using the distance formula:
Find the slope of the line passing through the points:
Use the slope and one of the points to write the equation in point-slope form, then rearrange to slope-intercept form: Then solve for to get .
To find the y-intercept, set in your equation and solve for .

Try solving on your own before revealing the answer!

Final Answer:

Midpoint:
Distance: $50$
Slope:
Equation:
Y-Intercept:
Graph: Plot the points and , draw the line through them, and mark the y-intercept at .

Each value was found by substituting the coordinates into the respective formulas. The equation is in slope-intercept form, and the graph should reflect all these features.

Q2. Given the equation , find the following: X-Intercept, Y-Intercept, Slope, Graph

Background

Topic: Linear Equations in Standard Form

This question tests your ability to extract key features (intercepts and slope) from a linear equation in standard form and to graph the line.

Key Terms and Formulas

Standard Form:
X-Intercept: Set and solve for
Y-Intercept: Set and solve for
Slope: (from )
Slope-Intercept Form:

Step-by-Step Guidance

To find the x-intercept, set in and solve for .
To find the y-intercept, set in and solve for .
To find the slope, rearrange the equation into slope-intercept form () or use .
Once you have the intercepts and slope, you can sketch the graph by plotting the intercepts and drawing the line with the calculated slope.

Try solving on your own before revealing the answer!

Final Answer:

X-Intercept:
Y-Intercept:
Slope:
Graph: Plot the intercepts and draw a straight line through them with the calculated slope.

Each intercept is found by setting the other variable to zero. The slope is negative because the coefficient of is positive and is positive in standard form.

Q3. Solve & Graph the following systems of linear equations:

Background

Topic: Systems of Linear Equations

This question tests your ability to solve systems of two linear equations using algebraic methods and to graph the solution.

Key Terms and Methods

System of Equations: Two or more equations with the same variables
Solving Methods: Substitution and Elimination
Solution: The point(s) where the lines intersect

Step-by-Step Guidance

Write each equation clearly. If the equations are not in standard form, rearrange them as needed.
Choose a method (substitution or elimination) to solve for one variable.
Once you have one variable, substitute back into one of the original equations to solve for the other variable.
Write the solution as an ordered pair .
To graph, plot both lines on the same coordinate plane and mark their intersection point.

Try solving on your own before revealing the answer!

Final Answer:

For the first system (assuming equations are and ): Solution:
For the second system (assuming equations are and ): First, . Substitute into the second equation to find . Solution:
Graph: Plot both lines for each system and mark the intersection point as the solution.

Each solution is found by algebraic manipulation and substitution. The intersection point represents the solution to the system.