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Chapter 0: Problem 13

Graph the equation by plotting points. $$y=-3 x+2$$

Short Answer

Expert verified
Graph the line through points (-1, 5), (0, 2), and (1, -1) by plotting and connecting them.

Step by step solution

01

Choose Values for x

We first choose several values for the variable \( x \) to calculate corresponding \( y \) values. Choosing simpler numbers can help plot the points easily on a graph. Let's pick \( x = -1, 0, 1 \).
02

Calculate Corresponding y Values

Substitute each chosen \( x \) value into the equation \( y = -3x + 2 \) to find \( y \).For \( x = -1 \):\( y = -3(-1) + 2 = 3 + 2 = 5 \).For \( x = 0 \):\( y = -3(0) + 2 = 0 + 2 = 2 \).For \( x = 1 \):\( y = -3(1) + 2 = -3 + 2 = -1 \).
03

Plot the Points

Plot these calculated points on the coordinate plane: \((-1, 5)\), \( (0, 2) \), and \( (1, -1) \). Each point corresponds to an ordered pair from the calculations.
04

Draw the Line

Draw a straight line through the plotted points \( (-1, 5), (0, 2), (1, -1) \). Since the equation is of the form \( y = mx + b \), it represents a straight line, and combined points should lie on this line.

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Key Concepts

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

Coordinate Plane
The coordinate plane is a two-dimensional surface that allows us to graph equations by assigning a pair of numerical coordinates. These coordinates, often written as
  • (x, y): where x represents the horizontal position and y the vertical position.
Each coordinate is based on the axes of the plane. The horizontal axis is called the x-axis and the vertical one is the y-axis. Where these two axes intersect is known as the origin, usually marked as (0, 0). Using the coordinate plane allows us to easily locate points corresponding to the solutions of equations. This is crucial when graphing linear equations, which becomes more intuitive by visualizing how the values connect on this grid.
Linear Equation
A linear equation is an algebraic equation of the form
  • y = mx + b
where m is the slope and b is the y-intercept. This equation defines a straight line on the coordinate plane. The reason it鈥檚 called "linear" is because the graph of the equation is a straight line.
Linear equations can describe various situations in life, from rooftop slopes to financial budgeting. By understanding this form, you can easily predict changes in the variable "y" as "x" moves along the line. This simple mathematical relationship lays the groundwork for more complex analyses in algebra.
Slope-Intercept Form
The slope-intercept form of a linear equation is particularly user-friendly for graphing. Given as
  • y = mx + b
it provides direct insight into crucial graph characteristics:
  • Slope (m): Indicates the steepness and direction of the line.
A positive m means the line ascends from left to right, while a negative m shows a descent.
  • Intercept (b): The point where the line crosses the y-axis.
Since
  • b = 2 in the equation y = -3x + 2, this line meets the y-axis at the point (0, 2).
Understanding this form not only simplifies graphing but also helps in quickly analyzing and interpreting linear relationships.

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Most popular questions from this chapter

Explain the mistake that is made. Find the \(x\) - and \(y\) -intercepts of the line with equation \(2 x-3 y=6\) Solution: \(x\) -intercept: set \(x=0\) and solve for \(y\) \(-3 y=6\) The \(x\) -intercept is (0,-2) \(y=-2\) \(y\) -intercept: set \(y=0\) and solve for \(x . \quad 2 x=6\) The \(y\) -intercept is (3,0) \(x=3\)

A speedboat takes 1 hour longer to go 24 miles up a river than to return. If the boat cruises at 10 mph in still water, what is the rate of the current?

Explain the mistake that is made. Solve the equation \(\frac{4}{p}-3=\frac{2}{5 p}\) Solution: \(\quad(p-3) 2=4(5 p)\) Cross multiply. $$ \begin{aligned} 2 p-6=& 20 p \\ -6=& 18 p \\ p=-& \frac{6}{18} \\ p=-\frac{1}{3} \end{aligned} $$ This is incorrect. What mistake was made?

Determine whether the lines are parallel, perpendicular, or neither, and then graph both lines in the same viewing screen using a graphing utility to confirm your answer. $$\begin{aligned} &y_{1}=17 x+22\\\ &y_{2}=-\frac{1}{17} x-13 \end{aligned}$$

Show that the sum of the roots of a quadratic equation is equal to \(-\frac{b}{a}\).
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