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Chapter 2: Problem 9

Find the value of \(t\) if \(m\) is the slope of the line that passes through the given points. a. \((3, t)\) and \((-2,1), m=-4\) b. (5,6) and \((t, 9), m=2\)

Short Answer

Expert verified
Part a: \( t = -19 \); Part b: \( t = 6.5 \).

Step by step solution

01

Understand the Slope Formula

The slope formula is given by \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] where \( (x_1, y_1) \) and \( (x_2, y_2) \) are the coordinates of the two points, and \( m \) is the slope.
02

Apply the Slope Formula to Part a

For part a, the points are \((3, t)\) and \((-2, 1)\) and the slope \( m = -4 \). We substitute these into the slope formula: \[-4 = \frac{1 - t}{-2 - 3} \] Simplify the denominator: \[-4 = \frac{1 - t}{-5} \]
03

Solve for t in Part a

To solve for \( t \), multiply both sides by \(-5\): \(-4 \times -5 = 1 - t \) Simplify: \[ 20 = 1 - t \] Rearrange to solve for \( t \): \[ t = 1 - 20 \] Thus, \[ t = -19 \]
04

Apply the Slope Formula to Part b

For part b, the points are \((5, 6)\) and \((t, 9)\), and the slope \( m = 2 \). Substitute these into the slope formula: \[ 2 = \frac{9 - 6}{t - 5} \] Simplify the numerator: \[ 2 = \frac{3}{t - 5} \]
05

Solve for t in Part b

To solve for \( t \), multiply both sides by \( t - 5 \): \[ 2(t - 5) = 3 \] Distribute the 2: \[ 2t - 10 = 3 \] Add 10 to both sides and then solve for \( t \): \[ 2t = 13 \] \[ t = \frac{13}{2} \] Thus, \[ t = 6.5 \]

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Key Concepts

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

slope formula
The slope formula is a fundamental concept in algebra and coordinate geometry. It helps us find the steepness or incline of a line connecting two points. The formula for the slope (denoted as \( m \)) is \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]. Here, \( (x_1, y_1) \) and \( (x_2, y_2) \) are the coordinates of the two points on the line. The slope indicates the rate of change between the \( y \)-coordinates with respect to the \( x \)-coordinates.

A few important points to remember about the slope:
  • If the slope is positive, the line ascends from left to right.
  • If the slope is negative, the line descends from left to right.
  • A zero slope means the line is horizontal.
  • An undefined slope (division by zero) means the line is vertical.
Understanding these basics will help you solve many problems involving linear equations and coordinate geometry.
solving linear equations
Solving linear equations is an essential skill in algebra. When given an equation involving a variable, the goal is to solve for that variable. This often involves using arithmetic operations such as addition, subtraction, multiplication, and division.

Let's look at examples based on the slope formula:
  • In part (a) of the exercise, we had to solve \( -4 = \frac{1 - t}{-5} \). Multiplying both sides by \( -5 \) gives \( 20 = 1 - t \). Then, rearranging, we find \( t = -19 \).
  • In part (b), we had \( 2 = \frac{3}{t-5} \). Multiplying both sides by \( t-5 \), results in \( 2(t-5) = 3 \). Distributing and solving for \( t \), we get \( t = 6.5 \).

Steps for solving linear equations:
  • Isolate the variable by performing inverse operations.
  • Simplify expressions as you perform these operations.
  • Check your solution by substituting it back into the original equation.
With practice, solving these equations becomes more intuitive.
coordinate geometry
Coordinate geometry, also known as analytic geometry, is the study of geometry using a coordinate system. This branch of mathematics allows us to describe geometric figures and their properties using algebraic equations.

The basics of coordinate geometry involve understanding points, lines, and planes in a coordinate system. Each point is given by an ordered pair \( (x, y) \) in a 2D space. Lines can be described using linear equations of the form \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept.

Key points to remember:
  • The slope formula connects algebraic and geometric concepts by quantifying the steepness of a line between two points.
  • Understanding how to use and manipulate coordinates is crucial for solving many real-world problems.
  • Visualizing these concepts on a graph can aid comprehension, as you can see how changes in equations reflect changes in geometry.
Coordinate geometry forms a bridge between algebra and geometry, making it an indispensable tool for students.

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Most popular questions from this chapter

On the same axes, graph (and label with the correct equation) three lines that go through the point (0,2) and have the following slopes: a. \(m=-\frac{1}{2}\) b. \(m=-2\) c. \(m=-\frac{5}{6}\)

Construct an equation and sketch the graph of its line with the given slope, \(m,\) and vertical intercept, \(b .\) (Hint: Find two points on the line.) a. \(m=2, b=-3\) b. \(m=-\frac{3}{4}, b=1\) c. \(m=0, b=50\)

Your favorite aunt put money in a savings account for you. The account earns simple interest; that is, it increases by a fixed amount each year. After 2 years your account has \(\$ 8250\) in it and after 5 years it has \(\$ 9375 .\) a. Construct an equation to model the amount of money in your account. b. How much did your aunt put in initially? c. How much will your account have after 10 years?

Assume two individuals, \(A\) and \(B\), are traveling by car and initially are 400 miles apart. They travel toward each other, pass and then continue on. a. If \(A\) is traveling at 60 miles per hour, and \(B\) is traveling at \(40 \mathrm{mph},\) write two functions, \(d_{\mathrm{A}}(t)\) and \(d_{\mathrm{B}}(t),\) that describe the distance (in miles) that \(\mathrm{A}\) and \(\mathrm{B}\) each has traveled over time \(t\) (in hours). b. Now construct a function for the distance \(D_{\mathrm{AB}}(t)\) between A and \(B\) at time \(t\) (in hours). Graph the function for \(0 \leq t \leq 8\) hours.

The gas gauge on your car is broken, but you know that the car averages 22 miles per gallon. You fill your 15.5 -gallon gas tank and tell your friend, 'I can travel 300 miles before 1 need to fill up the tank again." Explain why this is true,
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