Problem 51
Question
Write an equation of the line that passes through the points. Use the slope- intercept form (if possible). If not possible, explain why and use the general form. Use a graphing utility to graph the line (if possible). $$(5,-1),(-5,5)$$
Step-by-Step Solution
Verified Answer
The equation of the line that passes through the points (5,-1) and (-5,5) is \(y = -0.6x + 2\).
1Step 1: Calculate the Slope
The slope \(m\) is calculated using the formula \(m = (y_2 - y_1) / (x_2 - x_1)\). Here, \(x_1 = 5\), \(y_1 = -1\), \(x_2 = -5\) and \(y_2 = 5\). Plugging these into the formula gives us \(m = (5 - (-1)) / ((-5) - 5) = -6/10 = -0.6\).
2Step 2: Use the Slope-Intercept Form for the Equation
We then use any of the given points and the slope in the equation \(y = mx + b\) to find the y-intercept. Let's choose the point (5,-1). Substituting \(m = -0.6\), \(x = 5\) and \(y = -1\) into the equation we get: -1 = -0.6*5 + b. Solving this equation we get \(b = 2\).
3Step 3: Write the Full Equation of the Line
The full equation of the line in slope-intercept form is \(y = -0.6x + 2\).
Key Concepts
Slope-Intercept FormCalculating SlopeGraphing LinesLinear Equations
Slope-Intercept Form
The slope-intercept form of a linear equation is an intuitive way to describe a line. This form is written as \( y = mx + b \) , where \( m \) represents the slope of the line and \( b \) represents the y-intercept, which is the point where the line crosses the y-axis.
In basic terms, the slope \( m \) describes how steep the line is, and the y-intercept \( b \) tells us where the line starts on the y-axis. If you understand two things—a measure of steepness and a starting point—you can draw a straight line across a two-dimensional plane. This form is favored for its simplicity and ease of graphing.
In basic terms, the slope \( m \) describes how steep the line is, and the y-intercept \( b \) tells us where the line starts on the y-axis. If you understand two things—a measure of steepness and a starting point—you can draw a straight line across a two-dimensional plane. This form is favored for its simplicity and ease of graphing.
Calculating Slope
The slope of a line measures the rate at which \( y \) value changes for a corresponding change in the \( x \) value. It's a number that describes the tilt or gradient of the line. To calculate the slope, represented as \( m \) , you use two points on the line with coordinates \( (x_1, y_1) \) and \( (x_2, y_2) \) . The formula is \( m = (y_2 - y_1) / (x_2 - x_1) \) .
A positive slope indicates that the line tilts upwards as we move from left to right, while a negative slope tilts downwards. If the slope is zero, the line is horizontal, and if the slope is undefined, the line is vertical. Calculating the slope is a fundamental skill in algebra that is used for graphing lines and solving linear equations.
A positive slope indicates that the line tilts upwards as we move from left to right, while a negative slope tilts downwards. If the slope is zero, the line is horizontal, and if the slope is undefined, the line is vertical. Calculating the slope is a fundamental skill in algebra that is used for graphing lines and solving linear equations.
Graphing Lines
Graphing lines involves plotting a line on a coordinate plane based on an equation. With the slope-intercept form, graphing becomes a two-step process: first, plot the y-intercept \( b \) on the y-axis. Then use the slope \( m \) to determine the tilt of the line. If the slope is a fraction, it can be interpreted as 'rise over run', or the vertical change over the horizontal change between any two points on the line.
With technology, graphing becomes even easier. You can use graphing calculators or graphing software to input your linear equation and visually represent it instantly. This not only saves time but helps in visual learning, making the abstract concept of algebra more concrete.
With technology, graphing becomes even easier. You can use graphing calculators or graphing software to input your linear equation and visually represent it instantly. This not only saves time but helps in visual learning, making the abstract concept of algebra more concrete.
Linear Equations
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 create straight lines when graphed on a coordinate plane. The standard form of a linear equation is \( Ax + By = C \) , but there are other forms, such as the point-slope form and the slope-intercept form discussed earlier.
Understanding linear equations is fundamental to algebra. They represent relationships where there is a constant rate of change between variables. They can be used to model simple real-life situations, making them an essential part of mathematical education and an invaluable tool in various fields including science, engineering, and economics.
Understanding linear equations is fundamental to algebra. They represent relationships where there is a constant rate of change between variables. They can be used to model simple real-life situations, making them an essential part of mathematical education and an invaluable tool in various fields including science, engineering, and economics.
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