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Chapter 12: Problem 40

Find the equation of the line through the given points. $$(-3,9) \text { and }(1,1)$$

Short Answer

Expert verified
The equation of the line that passes through the points (-3,9) and (1,1) is \(y = -2x + 3\).

Step by step solution

01

Calculate the Slope

The slope of a line formed by two points \((x_1,y_1)\) and \((x_2,y_2)\) is given by the formula: \(m = \frac{y_2 - y_1}{x_2 - x_1}\). Using the given points, \((-3,9)\) and \((1,1)\), the slope \(m\) is calculated as : \(m = \frac{1 - 9}{1 - (-3)} = -2\).
02

Get the Equation of the Line

Then, the equation of the line can be found using the slope-intercept form \(y = mx + c\). Substituting in \(m = -2\) and using one of the points, let's say \((-3,9)\), we get: \(9 = -2(-3) + c\). Rearrange this to find \(c\).
03

Calculate the y-intercept

Solve for \(c\) in the equation \(9 = 6 + c\). This gives \(c = 3\).
04

Formulate the Line Equation

Now that we have \(m\) and \(c\), we can write the equation of the line as \(y = -2x + 3\).

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Key Concepts

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

Slope Calculation
When learning about linear equations, understanding how to calculate the slope is fundamental. The slope, often represented by the letter \(m\), describes how steep a line is. When given two points, such as \((-3, 9)\) and \((1, 1)\), it is possible to calculate the slope using the formula:
  • \(m = \frac{y_2 - y_1}{x_2 - x_1}\)
This means you subtract the \(y\)-coordinates of the two points and divide that by the difference in the \(x\)-coordinates.
The slope, in our example, is calculated as:
  • \(m = \frac{1 - 9}{1 - (-3)} = \frac{-8}{4} = -2\)
A negative slope means the line goes down as it moves from left to right. This calculation is simple but essential for understanding the behavior of the line.
Slope-Intercept Form
The slope-intercept form is a straightforward way to describe a line with the equation \(y = mx + c\). This equation serves to represent a line graphically and is quite intuitive to use:
  • The \(m\) stands for the slope of the line.
  • The \(c\) (often called the "y-intercept") is where the line crosses the \(y\)-axis.
Once you know the slope, as in our earlier example \(m = -2\), and one point on the line, you can substitute these values into the slope-intercept form.
For instance, using the point \((-3, 9)\), substitute into the equation:
  • \(9 = -2(-3) + c\)
  • Simplifying this, \(9 = 6 + c\)
  • It follows that \(c = 3\)
Now, the equation of the line is \(y = -2x + 3\), which means the slope is \(-2\) and it crosses the \(y\)-axis at \(3\).
Coordinate Geometry
Coordinate geometry, or analytic geometry, merges algebra with geometry by using the coordinate plane. It handles the study of geometric figures using their algebraic equations. When dealing with problems like finding the equation of a line, coordinate geometry provides a framework to visualize solutions.
  • The X-axis and Y-axis divide the plane into four quadrants.
  • Points are represented by coordinates \((x, y)\) which denote their positions on this plane.
  • The equation of a line provides a precise description of all points, \((x, y)\), lying on the line.
Through coordinate geometry, equations such as \(y = -2x + 3\) give simple and complete descriptions of lines. It's an essential part of understanding mathematical relationships and visualizing them in real-world applications. This allows students to clearly see how lines can rise, fall, and intersect with various axes as described by their algebraic equations.

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Most popular questions from this chapter

Graph by using the slope and \(y\) -intercept. (GRAPH CAN'T COPY) $$y=\frac{3}{4} x+1$$

Graph by using the slope and \(y\) -intercept. (GRAPH CAN'T COPY) $$y=-x-3$$

Determine whether the line through \(P_{1}\) and \(P_{2}\) is parallel, perpendicular, or neither parallel nor perpendicular to the line through \(Q_{1}\) and \(Q_{2}\). $$P_{1}(1,-1), P_{2}(3,-2) ; Q_{1}(-4,1), Q_{2}(2,-5)$$

Evaluate a function. Given \(f(x)=x^{2}-1,\) find \(f(1)\).

Suppose \(A\) is a negative number, and \(B\) and \(C\) are positive numbers. Does the \(y\) -intercept of the graph of \(A x+B y=C\) lie above or below the \(x\) -axis? Does the graph slant upward to the right or downward to the right?
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