Problem 10 - Algebra 1

Question

Write an equation of the line in point-slope form that passes through the given point and has the given slope. $$ (-2,-9), m=4 $$

Step-by-Step Solution

Answer

The point-slope form of the equation of the line that passes through the point (-2,-9) with a slope of 4 is \(y + 9 = 4(x + 2)\).

Step 1: Identify the given coordinates and slope

The given point is \((-2,-9)\) and the slope is \(4\). Here, \(x1 = -2\), \(y1 = -9\) and \(m = 4\).

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

The formula for the point-slope equation of a line is \(y - y1 = m(x - x1)\). Substitute the values of \(x1\), \(y1\) and \(m\) into this formula: \(y - (-9) = 4(x - (-2))\).

Step 3: Simplify the equation

Simplify the equation derived in Step 2. The equation simplifies to \(y + 9 = 4(x + 2)\).

Key Concepts

Linear Equations

Linear equations form the foundation of algebra and represent relationships between two variables as a straight line when plotted on a graph. These equations follow the general format of ax + by = c, where a, b, and c are constants. To visualize this concept, imagine you're plotting points on a grid. Each solution to the equation is a point that lies on the same straight line, revealing a fixed rate of change between variables.

Understanding linear equations is key as they model real-world phenomena like motion at constant speed or currency conversion rates. When we refer to the equation of a line, we often use different forms such as slope-intercept (y = mx + b) or point-slope form, which is crucial when we know a particular point on the line and its slope. The problem we're discussing uses the point-slope form due to such given information.

Algebraic Expressions

Algebraic expressions are like sentences in the language of mathematics, composed of variables, numbers, and arithmetic operations such as addition and subtraction. Unlike equations, expressions do not have an equals sign; they are not statements of equality but rather mathematical phrases that represent a quantity.

When tackling an algebraic expression, it's helpful to think about substituting variables with values, much like filling in the blanks in a sentence to change its meaning. This substitution is exactly what we're doing when solving our exercise: taking the known values of a point's coordinates and a line's slope to build an equation, which is essentially a more complex algebraic expression that reveals how two variables are related.

Slope of a Line

The slope of a line is a measure of its steepness, which is a crucial concept in algebra, as it quantifies the rate at which y changes with respect to x. Think of it as the incline of a hill: a larger slope means a steeper hill. Mathematically, it's calculated as the 'rise over run', or the change in y divided by the change in x between two points on the line.

The slope is typically denoted by m, and it can be positive, negative, zero, or undefined. A positive slope means the line rises as it moves from left to right, while a negative slope indicates it falls. A slope of zero is a flat line, and an undefined slope means the line is vertical. In our exercise, the slope is 4, signaling that for every unit increase in x, y rises by 4 units, leading to an upward sloping line.