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Chapter 3: Problem 2

Find the slope and one point on each line. $$y-(-2)=-4[x-(-1)]$$

Short Answer

Expert verified
The slope of the line is -4 and the point on the line is (-1, -2).

Step by step solution

01

Identification of the Slope

From the equation \(y-(-2)=-4[x-(-1)]\), we can see that the slope (m) can be found where -4 is in the equation. Hence, the slope (m) is -4.
02

Identification of the Point

Looking at the provided equation again, we can take notice of (-1, -2) where -1 correlates to x1 and -2 to y1. Therefore, the point is (-1, -2).

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Key Concepts

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

Linear Equations
Linear equations express relationships between variables with a consistent rate of change. They are called \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. These equations describe straight lines on a graph. Linear equations stand out because they show how two variables relate through addition, subtraction, multiplication, or division of constants and variables.
  • Slope-intercept form: This is a popular format, represented as \( y = mx + b \).
  • Standard form: This uses the formula \( Ax + By = C \), where \( A \), \( B \), and \( C \) are constants.
  • Point-slope form: Useful when you know a point and the slope, as in \( y - y_1 = m(x - x_1) \).
Recognizing different formats helps in solving problems and graphing their solutions.
Slope
The slope is a measure of how steep a line is. Symbolized by \( m \), it indicates how much \( y \) changes for a change in \( x \). Positive slopes rise from left to right, while negative slopes fall. A zero slope means a horizontal line, and undefined slopes mean a vertical line.To find the slope from a linear equation in point-slope form, isolate the component with \( m \). Here, from the step-by-step solution, \( y - (-2) = -4 [x - (-1)] \), the slope is \(-4\).
  • Consistent Change: The slope signifies the ratio of vertical change to horizontal change between any two points on a line.
  • Point-Slope Relationship: Captured in the formula \( y - y_1 = m(x - x_1) \), immediately reveals the slope value.
Understanding slope is crucial for analyzing trends and predicting future values in graphs.
Coordinates
Coordinates pinpoint the exact position of a point on a graph using an ordered pair \((x, y)\). They help identify specific points on the Cartesian coordinate system, which consists of horizontal (x-axis) and vertical (y-axis) lines.Each point on a plane corresponds to a unique pair of numbers, \( (x, y) \), making it easy to locate on a graph. For example, in the given exercise, the coordinates are \((-1, -2)\), derived directly from the equation.
  • Axis Reference: The x-coordinate shows horizontal position, while the y-coordinate shows vertical position.
  • Ordered Pair: Presented as \((x, y)\), they follow a convention with x-values written first.
Coordinates are fundamental for plotting points and constructing graphs. They provide a visual representation of the relationships within a linear equation.

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Most popular questions from this chapter

Set up a variation equation and solve for the requested value. The force of gravity acting on an object varies directly with the mass of the object. The force on a mass of 5 kilograms is 49 newtons. What is the force acting on a mass of 12 kilograms?

Set up a variation equation and solve for the requested value. The interest earned on a fixed amount of money varies jointly with the annual interest rate and the time that the money is left on deposit. If an account earns \(120\)dollar at \(8 \%\) annual interest when left on deposit for 2 years, how much interest would be earned in 3 years at an annual rate of \(12 \% ?\)

Express each combined variation as an equation. Then find the requested value. Assume that all variables represent positive numbers. SEE EXAMPLE 4. (OBJECTIVE 4) \(p\) varies directly with \(q\) and inversely with \(r .\) If \(p=5\) when \(q=1\) and \(r=6,\) find \(p\) when \(q=5\) and \(r=10\)

Solve each inequality and graph the solution. $$-x+2 \leq 5$$

Explain the meaning of combined variation.
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