Problem 45
Question
Write in point-slope form the equation of the line that passes through the given point and has the given slope. (Lesson 5.2) $$ (5,-3), m=-4 $$
Step-by-Step Solution
Verified Answer
The equation of the line that passes through the point (5,-3) and has a slope of -4 in the point-slope form is \(y = -4x + 17\).
1Step 1: Identifying the given values
The given points are \(x1=5\), \(y1=-3\) and the slope \(m=-4\). These values are to be substituted into the point-slope form equation to get the equation of the line.
2Step 2: Substituting the values
Replace \(m\), \(x1\), and \(y1\) in the equation \(y - y1 = m(x - x1)\) with the given values. This gives the equation: \(y - (-3) = -4 (x - 5)\).
3Step 3: Simplifying the equation
The equation can be simplified by performing the operations. This gives the equation: \(y + 3 = -4x + 20\). Further simplifying gets us the final equation of the line: \(y = -4x + 17\).
Key Concepts
Line EquationSlope of a LineAlgebraic Expressions
Line Equation
In the world of algebra, equations representing lines are a fundamental concept. A line equation is a mathematical expression that describes a straight line on a coordinate plane. Each line in two dimensions can be represented by an equation in various forms, including slope-intercept, point-slope, and standard form.
The point-slope form, in particular, is handy when we have a point on the line \( (x_1, y_1) \) and the slope of the line \( m \) known. It is written as \( y - y_1 = m(x - x_1) \). When we apply this formula, what we're doing is setting up a relationship that says the difference between any point's y-coordinate and \( y_1 \) will be proportional to the difference between that point's x-coordinate and \( x_1 \)—the proportion being the slope \( m \) of the line. In our exercise example, the given point \( (5, -3) \) and slope \( -4 \) yield the line equation in point-slope form.
The point-slope form, in particular, is handy when we have a point on the line \( (x_1, y_1) \) and the slope of the line \( m \) known. It is written as \( y - y_1 = m(x - x_1) \). When we apply this formula, what we're doing is setting up a relationship that says the difference between any point's y-coordinate and \( y_1 \) will be proportional to the difference between that point's x-coordinate and \( x_1 \)—the proportion being the slope \( m \) of the line. In our exercise example, the given point \( (5, -3) \) and slope \( -4 \) yield the line equation in point-slope form.
Slope of a Line
The slope of a line is a measure that captures the 'steepness' and the 'direction' of the line. Formally, slope is defined as the ratio of the vertical change (rise) to the horizontal change (run) between any two points on the line. The symbol \( m \) usually denotes slope in equations.
When the slope is positive, the line rises as it moves from left to right; when negative, it falls. A zero slope indicates a horizontal line, while an undefined slope (often resulting from a zero denominator) corresponds to a vertical line. In algebraic terms, if we take two points on the line, say \( (x_1, y_1) \) and \( (x_2, y_2) \), the slope \( m \) is calculated as \( m = \frac{y_2 - y_1}{x_2 - x_1} \). The exercise shows a negative slope \( m = -4 \), which suggests the line is downward sloping from left to right. Understanding the concept of slope allows you to visualize the orientation of the line even before plotting it.
When the slope is positive, the line rises as it moves from left to right; when negative, it falls. A zero slope indicates a horizontal line, while an undefined slope (often resulting from a zero denominator) corresponds to a vertical line. In algebraic terms, if we take two points on the line, say \( (x_1, y_1) \) and \( (x_2, y_2) \), the slope \( m \) is calculated as \( m = \frac{y_2 - y_1}{x_2 - x_1} \). The exercise shows a negative slope \( m = -4 \), which suggests the line is downward sloping from left to right. Understanding the concept of slope allows you to visualize the orientation of the line even before plotting it.
Algebraic Expressions
An algebraic expression is a combination of numbers, variables, and mathematical operators (addition, subtraction, multiplication, and division). Algebraic expressions encapsulate relationships and are used to model real-world scenarios mathematically. Unlike equations, expressions do not include an equality sign.
In our example, the point-slope form \( y - y_1 = m(x - x_1) \) can be considered as an algebraic expression because it contains variables \( (x, y) \) and constants \( x_1, y_1 \) and \( m \) with arithmetic operations in place. The expression becomes a full-fledged equation upon the introduction of a specific point and slope. Simplifying the expression, as shown in the exercise solution steps, often involves distributing multiplication across terms and combining like terms, leading to the simplified form of the line equation. Being comfortable with manipulating algebraic expressions is essential for solving a wide range of problems in algebra.
In our example, the point-slope form \( y - y_1 = m(x - x_1) \) can be considered as an algebraic expression because it contains variables \( (x, y) \) and constants \( x_1, y_1 \) and \( m \) with arithmetic operations in place. The expression becomes a full-fledged equation upon the introduction of a specific point and slope. Simplifying the expression, as shown in the exercise solution steps, often involves distributing multiplication across terms and combining like terms, leading to the simplified form of the line equation. Being comfortable with manipulating algebraic expressions is essential for solving a wide range of problems in algebra.
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Problem 45
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