Problem 11 - Elementary Algebra

Question

Find the equation of the line passing through the point (-2,9) having slope \(0 .\)

Step-by-Step Solution

Answer

Question: Find the equation of a line that passes through the point (-2, 9) and has a slope of 0. Answer: The equation of the line is y = 9.

Step 1: Recall the point-slope form equation

The point-slope form equation is given by: \(y - y_1 = m(x - x_1)\), where \((x_1, y_1)\) represents the given point and \(m\) is the slope.

Step 2: Plug in the given point and slope

In this case, the given point is \((-2, 9)\) and the slope \(m\) is \(0\). Therefore, we can rewrite the point-slope form equation as: \(y - 9 = 0(x - (-2))\)

Step 3: Simplify the equation

Now we will simplify the equation and solve for \(y\): \(y - 9 = 0(x + 2)\) \(y - 9 = 0\) \(y = 9\)

Step 4: Write the final equation

The equation of the line passing through the point \((-2, 9)\) having slope \(0\) is \(y = 9\).

Key Concepts

Equation of a Line

When dealing with the equation of a line, there are several forms it can take. One common form is the slope-intercept form, written as y = mx + b. This form quickly tells you the slope of the line, m, and where it intercepts the y-axis, known as the y-intercept, b.

Another versatile form is the point-slope form, which is particularly useful when you know a specific point the line passes through and the slope. This form is expressed as y - y_1 = m(x - x_1), where (x_1, y_1) is the point on the line and m is the slope. In the simplest case, when the slope m is zero, the equation simplifies to y = y_1, indicating a horizontal line. Understanding these equations and how to manipulate them algebraically is foundational for grasping linear relationships in algebra.

Slope of a Line

The slope of a line is a measure of its steepness and direction. Algebraically, it's the ratio of the vertical change (y or rise) to the horizontal change (x or run) between two points on the line. You can calculate it by using the formula m = (y_2 - y_1) / (x_2 - x_1). A positive slope means the line rises as it moves from left to right and a negative slope means it falls.

A slope of zero indicates a horizontal line, which is the concept applied in our exercise. This horizontal line will not 'rise' or 'fall' no matter how much you move to the left or right, so the equation reduces to simply y = constant, reflecting that the y-value remains constant at every point along the line.

Algebraic Equations

Algebraic equations are mathematical statements that use numerical and literal (using letters) components to represent a relationship. For instance, y = 2x + 3 is an algebraic equation that represents a linear relationship between x and y. By manipulating these equations, such as adding, subtracting, multiplying, or dividing by non-zero numbers, we can solve for unknowns or simplify expressions.

In the context of our exercise, we simplified a point-slope form equation to a much simpler form, y = 9. It’s these algebraic operations that make solving equations powerful, allowing us to express complex relationships in a manageable and understandable way.