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Chapter 13: Problem 17

line through \((1,1)\) and \((4,7)\)

Short Answer

Expert verified
The equation of the line is \(y = 2x - 1\).

Step by step solution

01

Identify the two points

Point 1 is \((1,1)\) and Point 2 is \((4,7)\)
02

Calculate the slope

The formula for the slope of a line connecting two points is \((y_2 - y_1)/(x_2 - x_1)\). Therefore, the slope \(m\) is \((7 - 1)/(4 - 1) = 2\)
03

Find the equation of the line

We can use the point slope form of the line equation to find the actual equation. The form is \(y - y_1 = m(x - x_1)\). Substituting \(m\) with 2 and \((x_1, y_1)\) with (1,1), the equation becomes \(y - 1 = 2(x - 1)\). Simplifying this equation, we find the equation of the line as \(y = 2x - 1\)

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Key Concepts

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

Slope Calculation
Calculating the slope of a line is a fundamental concept in understanding how lines behave on a graph. The slope describes the steepness and direction of a line. To find the slope between two points, like \((1, 1)\) and \((4, 7)\), follow these steps:
  • Identify the points: Here, we have \((x_1, y_1) = (1, 1)\) and \((x_2, y_2) = (4, 7)\).
  • Use the slope formula: \(m = \frac{y_2-y_1}{x_2-x_1}\).
  • Substitute the values: \(m = \frac{7-1}{4-1} = \frac{6}{3} = 2\).
The slope \(m = 2\) tells us the line rises 2 units for every 1 unit it goes right. This positive slope indicates the line is ascending as we move from left to right.
Point-Slope Form
The point-slope form is an essential tool for writing the equation of a line when you know a point on the line and its slope. The form is expressed as:
\[ y - y_1 = m(x - x_1) \]
This format is particularly handy if you have a specific point, \((x_1, y_1)\), and the slope \(m\). In our example, using \((1, 1)\) and \(m = 2\):
  • Plug into the formula: \(y - 1 = 2(x - 1)\).
  • Simplify the equation: \(y - 1 = 2x - 2\).
  • Rearrange to find \(y\): \(y = 2x - 1\).
This equation \(y = 2x - 1\) is the line's equation and can be used to find any point on the line.
Line through Two Points
Finding the line through two given points involves determining the slope and utilizing it to write the line's equation. Here鈥檚 a deeper look at the steps for the points \((1, 1)\) and \((4, 7)\):
  • First, calculate the slope using \(m = \frac{y_2-y_1}{x_2-x_1}\) which results in \(m = 2\).
  • Choose either of the points; typically, you鈥檒l start with the first point. In this case, \((1, 1)\).
  • Apply the point-slope form formula: \(y - y_1 = m(x - x_1)\).
  • Substitute the slope and one of the points into the formula: \(y - 1 = 2(x - 1)\).
Rearranging gives the equation \(y = 2x - 1\). This tells us all we need to define the relationship between the line and the points it's passing through.

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Most popular questions from this chapter

Suppose \(E\) is on \(\overline{P Q}\) and \(P E=\frac{1}{4} P Q .\) If \(P=\left(x_{1}, y_{1}\right)\) and \(Q=\left(x_{2}, y_{2}\right)\), $$\text { where } x_{1}

Refer to \(\triangle Q R S\) with vertices \(Q(-6,0), R(12,0),\) and \(S(0,12)\) a. Find the equations of the three perpendicular bisectors of the sides of \(\triangle Q R S\) b. Show that the three perpendicular bisectors meet in a point \(C\) (called the circumcenter). (See the hint from Exercise \(36(b) .\) ) c. Show that \(C\) is equidistant from \(Q, R,\) and \(S\) by using the distance formula. d. Find the equation of the circle that can be circumscribed about \(\triangle Q R S\).

Use coordinate geometry to prove each statement. First draw a figure and choose convenient axes and coordinates. The diagonals of a rectangle are congruent.

horizontal line through \((3,1)\)

Find each vector sum. Then illustrate each sum with a diagram like that on page 541. $$(-8,2)+(4,6)$$
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