Problem 66 - Algebra 1: Concepts and Skills

Question

Write in point-slope form the equation of the line that passes through the given point and has the given slope. $$ (0,-3), m=5 $$

Step-by-Step Solution

Answer

The equation of the line in point-slope form is \(y + 3 = 5x\).

Step 1: Identify the given coordinates and slope

From the problem, the given point is (0,-3) and the slope (\(m\)) is 5. So, \(x_1 = 0\), \(y_1 = -3\), and \(m = 5\).

Step 2: Substitute the values into the point-slope formula

Plugging \(x_1\), \(y_1\), and \(m\) into the point-slope form equation \(y - y_1 = m(x - x_1)\) , we get \(y - (-3) = 5(x - 0)\).

Step 3: Simplify the Equation

Simplifying the equation on both sides will give: \(y + 3 = 5x\). Therefore, the equation of the line in point-slope form that passes through the point (0,-3) and has slope 5 is \(y + 3 = 5x\).

Key Concepts

Understanding Linear Equations

Linear equations form the backbone of algebra and provide the means to describe the relationship between two variables in a straight line. These equations follow the general format of ax + by + c = 0, where a, b, and c are constants, and x and y represent the variables. The beauty of a linear equation lies in its simplicity and predictability, often graphed as a straight line on a coordinate plane.

In our exercise, the linear equation is represented in point-slope form, which is written as y - y_1 = m(x - x_1). Here, (x_1, y_1) is a specific point that the line passes through, and m represents the slope of the line. The point-slope form is particularly useful when you have one known point on the line and the slope. It allows you to write the equation of the line quickly and proceed with your algebraic manipulations or graphical analysis as needed.

Grasping the Slope of a Line

The slope of a line is a numerical measure of its steepness, typically represented by the letter m. It indicates how much the line rises or falls as you move horizontally from one point to another on the graph of a line. Formally, slope is defined as the ratio of the change in the y-coordinates (rise) to the change in the x-coordinates (run), which can be calculated when given two points (x_1, y_1) and (x_2, y_2) on the line: m = (y_2 - y_1) / (x_2 - x_1).

In our textbook example, the slope is simply given as m=5. This means for every one unit you move to the right along the x-axis, the line moves up by 5 units on the y-axis—a relatively steep line. Understanding the slope is crucial since it not only describes the tilt of the line but also allows us to use the point-slope form correctly to draft the equation of the line.

Deciphering Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and arithmetic operations such as addition, subtraction, multiplication, and division. Unlike equations, expressions don't have an equal sign; they're not complete thoughts. The expression represents a value that can change depending on the values substituted for its variables.

For instance, in the solution to our problem, the algebraic expression y + 3 = 5x is derived from the point-slope form. Simplification of these expressions is a fundamental skill in algebra. It involves combining like terms and using properties of arithmetic to rewrite expressions in a more manageable form. To get comfortable with algebraic expressions, practice is key—work through many different expressions and employ simplifying strategies efficiently to become proficient and prepared to move on to more complex algebraic concepts.