/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 54 (a) find the y-intercept. (b) ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 3: Problem 54

(a) find the y-intercept. (b) find the x-intercept. (c) find a third solution of the equation. (d) graph the equation. \(x+y=200\)

Short Answer

Expert verified
y-intercept: (0, 200), x-intercept: (200, 0), third solution: (100, 100).

Step by step solution

01

- Find the y-intercept

To find the y-intercept, set the value of x to 0 in the equation \(x + y = 200\). Therefore, the equation becomes \(0 + y = 200\). Solving for y, we get \(y = 200\). Thus, the y-intercept is \((0, 200)\).
02

- Find the x-intercept

To find the x-intercept, set the value of y to 0 in the equation \(x + y = 200\). Therefore, the equation becomes \(x + 0 = 200\). Solving for x, we get \(x = 200\). Thus, the x-intercept is \((200, 0)\).
03

- Find a third solution

To find a third solution, choose any value for x (other than 0 or 200). Let鈥檚 choose \(x = 100\). Substitute \(x = 100\) into the equation: \(100 + y = 200\). Solving for y, we get \(y = 100\). Thus, a third solution is \((100, 100)\).
04

- Graph the equation

To graph the equation \(x + y = 200\), plot the intercepts \((0, 200)\) and \((200, 0)\) on the coordinate plane. Draw a straight line through these points, showing all the solutions to the equation. Include the third solution \((100, 100)\) to confirm the accuracy of the line.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

y-intercept
The y-intercept is where a line crosses the y-axis on a graph. To find the y-intercept of a linear equation, set the value of x to 0 and solve for y. For example, with the equation \(x + y = 200\), setting x to 0 simplifies the equation to \(0 + y = 200\). Solving this, we get \(y = 200\). This tells us that the y-intercept is \(0, 200\). This point is crucial because it helps in quickly sketching the graph of the line.

Understanding the y-intercept:
  • It's the point where the line meets the y-axis.
  • Think of the y-intercept as the starting point when x is zero.
  • It provides one of the key points needed to draw the graph of the equation.
x-intercept
Similar to the y-intercept, the x-intercept is where a line crosses the x-axis. To find this point, we set y to 0 in the equation and solve for x. For example, in the equation \(x + y = 200\), setting y to 0 simplifies it to \(x + 0 = 200\). Solving this, we get \(x = 200\). Thus, the x-intercept is \(200, 0\). This point is essential for plotting the graph of the line.

Understanding the x-intercept:
  • It's the point where the line meets the x-axis.
  • It helps in understanding what the value of x is when y is zero.
  • Like the y-intercept, the x-intercept helps in graphing the equation.
graphing equations
Graphing linear equations is about plotting points on a coordinate plane and connecting them to form a line. For the equation \(x + y = 200\), you first find the intercepts. The y-intercept is \(0, 200\) and the x-intercept is \(200, 0\). Plot these intercepts on a graph.

Next, draw a straight line through these intercepts. This line represents all possible solutions to the equation \(x + y = 200\).

When graphing equations:
  • Always find and plot the intercepts first.
  • Choose additional points if necessary to ensure the line is accurate. For instance, choosing \(x = 100\) gives another solution point \(100, 100\).
  • Make sure the line extends through the entire graph and shows both positive and negative solutions.
finding solutions
Finding solutions to a linear equation means identifying pairs of x and y values that satisfy the equation. Beyond just intercepts, any point on the line of the graphed equation is a solution. For \(x + y = 200\), we already know that \(0, 200\) and \(200, 0\) are solutions. To find a third solution, you can choose any x-value and solve for y. For example, with \(x = 100\), solving \(100 + y = 200\) gives \(y = 100\), so \(100, 100\) is another solution.

Steps to Find Solutions:
  • Choose an arbitrary value for x or y.
  • Substitute this chosen value into the equation and solve for the other variable.
  • The resulting x and y values form a pair that is a solution.
Understanding this process helps cross-verify the plotted graph and ensures all possible solutions are accounted for.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

(a) graph the given points, and draw a line through the points. (b) use the graph to find the slope of the line. (c) use the slope formula to find the slope of the line. \((-20,-50) ;(10,30)\)

(a) find the \(y\)-intercept. (b) find the \(x\)-intercept. (c) use the slope formula to find the slope of the line. \(-8 x+3 y=48\)

Use a graphing calculator to graph each equation. Choose a window that shows the \(x\)-intercept and \(y\)-intercept. Sketch the graph; describe the window. \(y=-4\)

(a) represent the information as two ordered pairs. (b) find the average rate of change, \(m\). The estimated 12-month total service revenues for wireless service in the United States increased from \(\$ 113,538,221\) in 2005 to \(\$ 159,929,648\) in 2010 . Round to the nearest thousand. (Source: www.ctia.org)

The completed problem has one mistake. (a) Describe the mistake in words, or copy down the whole problem and highlight or circle the mistake. (b) Do the problem correctly. Problem: Use the slope formula to find the slope of the line that passes through \((6,2)\) and \((6,7)\). $$ \text { Incorrect Answer: } \begin{aligned} m &=\frac{y_{2}-y_{1}}{x_{2}-x_{1}} \\ m &=\frac{6-6}{7-2} \\ m &=\frac{0}{5} \\ m &=0 \end{aligned} $$
See all solutions

Recommended explanations on Math Textbooks

Applied Mathematics

Read Explanation

Statistics

Read Explanation

Calculus

Read Explanation

Pure Maths

Read Explanation

Geometry

Read Explanation

Decision Maths

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.