Problem 17 - Algebra 1: Concepts and Skills

Question

Write in point-slope form the equation of the line that passes through the given points. $$ (-4,5) \text { and }(4,5) $$

Step-by-Step Solution

Answer

The equation of the line in point-slope form that passes through the points (-4,5) and (4,5) is \(y = 5\).

Step 1: Calculate the Slope

The slope of a line passing through points \((x_1, y_1)\) and \((x_2, y_2)\) is calculated as \(m = (y_2 - y_1) / (x_2 - x_1)\). For the points (-4, 5) and (4, 5), this is \(m = (5 - 5) / (4 - (-4)) = 0\). So the slope of the line is 0.

Step 2: Substituting into Point-Slope Form

Using the point-slope form \(y - y_1 = m(x - x_1)\) and substituting m=0 and coordinates of any one of the points, for example (-4,5), the equation becomes \(y - 5 = 0 * (x + 4)\).

Step 3: Simplify the Expression

Simplify the equation to its simplest form to get the final equation. This leads to \(y - 5 = 0\), and further simplifying we get the final equation to be \(y = 5\).

Key Concepts

Slope Calculation

Understanding the concept of a slope is fundamental in algebra, particularly when dealing with linear equations. The slope is a measure of how steep a line is. Mathematically, it's the rate at which the y-coordinate of a point on the line changes relative to the x-coordinate. The formula for calculating a slope from two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is \( m = (y_2 - y_1) / (x_2 - x_1) \). When using this formula, if the result is positive, the line slopes upwards, and a negative result indicates it slopes downwards. A special case arises when the slope is zero, indicating a horizontal line, or undefined, showing a vertical line.

Looking at our example with points \( (-4, 5) \) and \( (4, 5) \) we found the slope \( m \) to be zero after applying the formula. This immediately informs us that our line is horizontal, which will be reflected in the final equation. To gain a strong grasp of linear equations, practicing slope calculation with various pairs of points is highly recommended.

Linear Equations

The linear equation is one of the most important concepts in algebra. A linear equation represents a straight line when graphed on a coordinate plane. Every linear equation is composed of variables, coefficients, and constants that express a relationship where the variables are only to the first power and do not multiply each other. Point-slope form is a specific type of linear equation given as \( y - y_1 = m(x - x_1) \) where \( m \) is the slope of the line and \( (x_1, y_1) \) is a point on the line.

Converting linear equations between different forms, such as point-slope, slope-intercept (\( y = mx + b \) ), and standard form (\( Ax + By = C \) ), is a skill that can offer deeper insight into the behavior of lines. In our example, since the slope is zero, the point-slope form simplifies to the equation \( y = 5 \) - a horizontal line with a y-intercept at 5. Linear equations describe a wealth of real-world phenomena, so their applications extend far beyond the classroom.

Algebraic Expressions

Algebraic expressions are combinations of letters and numbers using mathematical operations. These letters, often called variables, can represent unknown values or quantities that can change. For instance, in the point-slope equation \( y - y_1 = m(x - x_1) \), \( m \) and \( (x_1, y_1) \) are representations of the slope and a specific point on the line, respectively.

The process of simplifying algebraic expressions is important, as it makes equations easier to understand and solve. Simplification might involve combining like terms, using the distributive property, and removing parentheses. When we simplified the algebraic expression in our original problem to \( y = 5 \) from \( y - 5 = 0 \), we demonstrated this process. Simplification is a technique that aids in the visualization of equations, making algebra more accessible and applicable.