Problem 28 - Algebra 1: Concepts and Skills

Question

Write in point-slope form the equation of the line. Then rewrite the equation in slope-intercept form. $$ (5,-12), m=-11 $$

Step-by-Step Solution

Answer

The point-slope form of the line is \(y + 12 = -11x + 55\). The slope-intercept form of the line is \(y = -11x + 43\).

Step 1: Write the Point-Slope Equation

Substitute the values of the given point (5,-12) and slope m=-11 into the point-slope form equation \(y - y1 = m(x - x1)\). Here, \(y1\)=-12, \(x1\)=5 and \(m\)=-11. So the equation becomes \(y - (-12) = -11(x - 5)\).

Step 2: Simplify the Point-Slope Equation

The equation simplifies to \(y + 12 = -11x + 55\). This is the point-slope equation of the line.

Step 3: Rewrite in Slope-Intercept Form

To convert the point-slope equation into slope-intercept form, isolate \(y\) by subtracting 12 from both sides of the equation. So, the equation \(y + 12 = -11x + 55\) becomes \(y = -11x + 55 - 12\).

Step 4: Simplify the Slope-Intercept Equation

Simplify the equation to \(y = -11x + 43\). This is the slope-intercept form of the line.

Key Concepts

Slope-Intercept Form

The slope-intercept form of a linear equation is a simple and direct way to express the equation of a line. This form is given by the formula \(y = mx + b\), where \(m\) represents the slope of the line, and \(b\) represents the y-intercept—the point where the line crosses the y-axis.
Understanding this form allows us to quickly identify the steepness and position of a line on a graph.Let's break it down:

• Slope (\(m\)): Indicates the incline of the line, or how much the line rises or falls as you move along it. A positive slope means the line goes up, and a negative slope means it goes down.
• Y-Intercept (\(b\)): The y-coordinate where the line intersects the y-axis. This is where \(x\) equals 0.

Using the slope-intercept form makes it easy to graph a line and interpret its key features.

Linear Equations

Linear equations are equations of the first degree, meaning they have no exponents higher than one. These equations form straight lines when graphed, which is why they are called "linear."Key components include:

• Variables: Usually represented by \(x\) and \(y\), these are the unknowns or quantities we solve for.
• Coefficients: Numbers multiplied by the variables, indicating the slope and intercept in contextual scenarios.
• Constant Terms: Fixed numbers in the equation, often representing shifts in the line on the graph.

These equations can be written in many forms, such as slope-intercept (\(y = mx + b\)) or point-slope (\(y - y_1 = m(x - x_1)\)), each providing different insights into the line's properties. Understanding these equations helps in predicting and modeling real-world phenomena.

Algebra

Algebra is a branch of mathematics dealing with symbols and the rules for manipulating those symbols. It is the foundation for understanding linear equations and their forms. In algebra, we often:

• Use Variables to represent unknown numbers.
• Apply Operations: addition, subtraction, multiplication, and division to solve equations.
• Balance Equations by performing the same operation on both sides to isolate variables.

Understanding algebra is crucial for solving problems involving linear equations. It allows us to express complex real-world relationships in a mathematical form, providing clarity and precision.