Problem 1 - A Graphical Approach to College Algebra

Question

Write the slope-intercept form of the line that passes through the given point with slope \(m .\) Do not use a calculator. Through \((1,3), m=-2\)

Step-by-Step Solution

Answer

The equation is \( y = -2x + 5 \).

Step 1: Recall the Slope-Intercept Form

The slope-intercept form of a line is given by the equation \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept.

Step 2: Substitute the Slope

We know the slope \( m = -2 \). Substitute \( m = -2 \) into the equation, resulting in \( y = -2x + b \).

Step 3: Use the Given Point for Substitution

To find \( b \), use the given point \((1, 3)\). Substitute \( x = 1 \) and \( y = 3 \) into the equation: \( 3 = -2(1) + b \).

Step 4: Solve for the Y-Intercept (b)

Simplify the equation: \( 3 = -2 + b \). Solve for \( b \) by adding 2 to both sides: \( b = 5 \).

Step 5: Write the Final Equation

Substitute \( b = 5 \) back into the slope-intercept form. The equation becomes \( y = -2x + 5 \).

Key Concepts

Linear Equations

Linear equations are a fundamental part of algebra, describing a straight line on a graph. They take the form of \( y = mx + b \), known as the slope-intercept form. This equation helps us visualize how changes in the variables affect each other.

• They represent a constant rate of change between two variables.
• Each linear equation can be graphed as a straight line.

In any linear equation, \( y \) is the dependent variable, \( x \) is the independent variable, \( m \) represents the slope, and \( b \) is the y-intercept. Linear equations are pervasive in different areas, such as physics for motion equations, economics for supply and demand curves, and everyday tasks like budgeting.

Slope

The slope is a crucial component of a linear equation. It indicates how steep a line is and is noted as \( m \) in the slope-intercept form \( y = mx + b \).

• The slope tells us how much \( y \) changes for a one-unit change in \( x \).
• It's calculated as the 'rise' over 'run', where 'rise' refers to the vertical change and 'run' refers to the horizontal change.

A positive slope means the line rises as you move along the x-axis, whereas a negative slope means it falls. In our specific problem, the slope is \(-2\), which means for every unit increase in \( x \), \( y \) decreases by 2 units.

Y-Intercept

The y-intercept is where the line crosses the y-axis. It's denoted by \( b \) in the equation \( y = mx + b \).

• The y-intercept is the value of \( y \) when \( x \) is zero.
• It provides a starting point for drawing the graph of the linear equation.

In our exercise, through substituting the point \((1, 3)\) into the equation \( y = -2x + b \), we solved for \( b \) and found it to be 5. Hence, the y-intercept is 5, indicating that the line hits the y-axis at (0,5). Understanding the y-intercept is key to graphing and interpreting linear equations.

Algebra

Algebra is a branch of mathematics dealing with numbers and operations in symbolic form. It lays the groundwork for understanding linear equations. Algebra involves:

• Understanding and manipulating expressions and equations.
• Substituting known values into equations to find unknowns.

In our exercise, knowing algebra allows us to replace the given values, like the slope and coordinates, into the slope-intercept formula. This is how we ultimately derive \( y = -2x + 5 \). Algebra is powerful as it prepares students for advanced mathematics and applications in various scientific fields by teaching logical thinking and problem-solving.