Problem 69
Question
Write in point-slope form the equation of the line that passes through the given point and has the given slope. $$ (-1,7), m=-6 $$
Step-by-Step Solution
Verified Answer
The point-slope form of the line that passes through the point (-1,7) and has a slope of -6 is \(y - 7 = -6x - 6\).
1Step 1: Identify the known values
In this problem, the given point is \((-1,7)\) and the given slope \(m\) is \(-6\). Therefore, \(x1 = -1\), \(y1 = 7\), and \(m = -6\).
2Step 2: Substitute the known values into the point-slope formula
Now, substitute the given values into the point-slope formula, \(y - y1 = m(x - x1)\). The equation becomes \(y - 7 = -6(x - (-1))\).
3Step 3: Solve and simplify the equation
After substituting the known values, simplify the parenthesis in the equation. The equation becomes \(y - 7 = -6(x + 1)\). Then you can distribute the \(-6\) to get the final point-slope form of the line: \(y - 7 = -6x - 6\).
Key Concepts
Linear EquationsSlope of a LineAlgebraic Expressions
Linear Equations
Understanding linear equations is fundamental in algebra and represents a major component of the curriculum. A linear equation is an algebraic equation in which each term is either a constant or the product of a constant and a single variable. Linear equations can always be graphed as straight lines, hence the name 'linear'.
They take the form of \(y = mx + b\), where \(m\) represents the slope and \(b\) represents the y-intercept, the point where the line crosses the y-axis. The slope \(m\) indicates how steep the line is, and \(b\) gives the specific point at which the line hits the y-axis. In the exercise, we focused on a specific form of the linear equation, the point-slope form, which is particularly useful when you have a point and a slope and you want to quickly write down the equation of the line. The point-slope form is expressed as \(y - y1 = m(x - x1)\), where \(x1\), \(y1\) are the coordinates of the given point and \(m\) is the slope of the line.
They take the form of \(y = mx + b\), where \(m\) represents the slope and \(b\) represents the y-intercept, the point where the line crosses the y-axis. The slope \(m\) indicates how steep the line is, and \(b\) gives the specific point at which the line hits the y-axis. In the exercise, we focused on a specific form of the linear equation, the point-slope form, which is particularly useful when you have a point and a slope and you want to quickly write down the equation of the line. The point-slope form is expressed as \(y - y1 = m(x - x1)\), where \(x1\), \(y1\) are the coordinates of the given point and \(m\) is the slope of the line.
Slope of a Line
The slope of a line is a measure of its steepness and direction. In algebra, the slope is typically denoted as \(m\), and it is calculated as the ratio of the rise (the change in y) over the run (the change in x) between two distinct points on the line. It's essentially an expression of the vertical change divided by the horizontal change between those points, and can be positive, negative, zero, or undefined.
In our example, the slope is given as \(m = -6\). This negative value indicates that as one moves to the right along the x-axis, the line falls, which reflects a negative slope. When graphing, if the line moves from the bottom left to the top right, it is said to have a positive slope; conversely, a line moving from the top left to the bottom right, such as our example, has a negative slope. Understanding the concept of slope is crucial because it allows us to describe the direction and steepness of a line succinctly.
In our example, the slope is given as \(m = -6\). This negative value indicates that as one moves to the right along the x-axis, the line falls, which reflects a negative slope. When graphing, if the line moves from the bottom left to the top right, it is said to have a positive slope; conversely, a line moving from the top left to the bottom right, such as our example, has a negative slope. Understanding the concept of slope is crucial because it allows us to describe the direction and steepness of a line succinctly.
Algebraic Expressions
An algebraic expression is a mathematical phrase that can include numbers, variables, and operation symbols. Variables are symbols (like \(x\) or \(y\)) that stand in for unknown values or values that can change. In the context of our linear equation exercise, the equation \(y - 7 = -6(x + 1)\) is an algebraic expression representing a line.
Simplifying algebraic expressions is an essential skill. It often involves combining like terms and expanding multiplication to remove parentheses. For instance, the point-slope form algebraic expression we dealt with simplifies to eliminate the parentheses, making it easier to graph or convert into other forms such as slope-intercept form. Algebraic expressions are the foundation of most algebraic problems, so mastering their manipulation is key to success in algebra.
Simplifying algebraic expressions is an essential skill. It often involves combining like terms and expanding multiplication to remove parentheses. For instance, the point-slope form algebraic expression we dealt with simplifies to eliminate the parentheses, making it easier to graph or convert into other forms such as slope-intercept form. Algebraic expressions are the foundation of most algebraic problems, so mastering their manipulation is key to success in algebra.
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