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Chapter 0: Problem 30

Find an equation of the line: with \(m=-3,\) containing (5,-2).

Short Answer

Expert verified
The equation of the line is \(y = -3x + 13\).

Step by step solution

01

Write Down the Point-Slope Formula

The point-slope form of a linear equation is given by \(y - y_1 = m(x - x_1)\), where \((x_1, y_1)\) is a point on the line and \(m\) is the slope. For this problem, the point is \((5, -2)\) and the slope is \(m = -3\).
02

Substitute the Given Values

Substitute \((x_1, y_1) = (5, -2)\) and \(m = -3\) into the point-slope formula to get:\[y - (-2) = -3(x - 5)\].
03

Simplify the Equation

Simplify the expression to make it more readable:1. Simplifying the left side: \(y + 2\).2. Distribute \(-3\) in the right side: \(-3(x - 5) = -3x + 15\).Thus, the equation becomes: \[y + 2 = -3x + 15\].
04

Solve for y

To write the equation in slope-intercept form \(y = mx + b\), solve for \(y\):1. Subtract 2 from both sides: \[y = -3x + 15 - 2\].2. Simplify: \[y = -3x + 13\].This is the equation of the line.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Point-Slope Form
The point-slope form of a linear equation is a handy way to write the equation of a line when you know one point on the line and the slope. The generic formula is \(y - y_1 = m(x - x_1)\). Here, \((x_1, y_1)\) is the known point and \(m\) is the slope. This form is quite useful because it lets you plug in numbers directly when you have a point and the slope. You instantly get a linear equation, which is a straight line. In our example, the point is \((5, -2)\) and the slope is \(-3\).
  • Substitute the point and slope into the formula: \(y - (-2) = -3(x - 5)\).
  • It clearly shows how the line behaves around that point, with the slope showing steepness and direction.
This makes it a very visual way to understand how a line moves on a graph, connecting the algebraic process to geometric intuition.
Slope-Intercept Form
Slope-intercept form is another popular way to express the equation of a line. Its formula \(y = mx + b\) is intuitive and simple. Here, \(m\) is the slope and \(b\) is the y-intercept, which is the point where the line crosses the y-axis. This form is particularly useful for graphing because you can immediately see the starting point (the y-intercept) and how the line rises or falls with the slope.
  • The slope tells us if the line goes up or down as we move from left to right.
  • The y-intercept gives a neat starting point for plotting on a graph.
In our example, after converting from point-slope form, we got \(y = -3x + 13\). This equation tells us the line has a downward slope of \(-3\), and it crosses the y-axis at \(13\). It's a systematic way of visualizing the line's behavior by just looking at the equation.
Gradient
The gradient, more commonly referred to as slope in the context of linear equations, indicates how steep a line is. It's a measure of the direction and steepness of a line. In mathematical terms, the slope \(m\) is the change in \(y\) divided by the change in \(x\), often illustrated as \(m = \frac{\Delta y}{\Delta x}\).
  • A positive slope means the line rises as it moves from left to right.
  • A negative slope means the line falls.
  • A slope of zero means the line is flat, like a resting position.
For the equation we worked with, the slope \(-3\) tells us the line falls, moving three units down for every unit it moves to the right. This helps in understanding how fast the line ascends or descends depending on the value of the slope. Gradient is a fundamental concept in linear algebra and crucial for interpreting linear relationships graphically.

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