Problem 15 - Algebra 1

Question

Write an equation of the line in slope-intercept form. The slope is \(2 ;\) the \(y\) -intercept is \(-1\)

Step-by-Step Solution

Answer

The equation of the line is \(y = 2x - 1\).

Step 1: Identify the slope and y-intercept

From the problem, it's given that the slope \(m = 2\) and the y-intercept \(b = -1\). The slope is the number that multiplies the \(x\) term in the equation, and the y-intercept is the constant term.

Step 2: Substitute values into the equation

Substitute \(m = 2\) and \(b = -1\) into the slope-intercept equation \(y = mx + b\).

Key Concepts

Linear Equations

Linear equations are a type of equation where the highest power of the variable, typically denoted as \(x\), is one. These equations form a straight line when graphed on a coordinate plane. The general form is

• \(ax + by = c\)
• where \(a\), \(b\), and \(c\) are constants.
• The most useful form for graphing is the slope-intercept form \(y = mx + b\).

This form directly reveals the slope and y-intercept, making it much easier to understand and graph linear relationships. Linear equations are fundamental in algebra, providing a basis for solving various real-world problems, like predicting expenses or understanding rates of change.

Slope

The slope of a linear equation measures how steep the line is. It is represented by \(m\) in the equation \(y = mx + b\). The slope describes the rate of change between the dependent variable \(y\) and the independent variable \(x\). For example:

• If the slope \(m = 2\), for every one unit increase in \(x\), \(y\) increases by 2 units.
• A positive slope means the line rises as you move from left to right.
• A negative slope means the line falls as you move from left to right.
• A zero slope is a horizontal line, indicating no change in \(y\).

Understanding slope is crucial for graphing lines and analyzing how different variables interact. It is widely used in calculating trends and making predictions.

Y-Intercept

The y-intercept is the point where the line crosses the y-axis in a graph of a linear equation. It is expressed as \(b\) in the slope-intercept equation \(y = mx + b\).

• In the given exercise, the y-intercept \(b = -1\).
• This means the line crosses the y-axis at \(-1\).
• It represents the value of \(y\) when \(x = 0\).
• The y-intercept helps to quickly draw the line once the slope is known.

By understanding the y-intercept, you can start graphing a line more easily since it gives you a starting point on the graph where the line splits the y-axis. It's a simple but powerful concept in understanding linear relationships.

Algebra Problems

Algebra problems often require reshaping and understanding equations to find unknown variables, which is a key skill in solving and interpreting relationships in math. Here's how to work through algebra problems involving linear equations:

Identify known elements: like slope \(m\) and y-intercept \(b\).

Substitute these known values into formulae: for example, into the slope-intercept form \(y = mx + b\).

Graph the equations or solve for other unknowns using algebraic manipulation.

Check work by substituting solutions back into the original equations.

Solving algebra problems with linear equations helps develop critical thinking and problem-solving skills. They are involved in diverse applications, from calculating distances to managing finances. Understanding how to work with these equations is crucial in mastering algebra and tackling more complex math topics.