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Chapter 0: Problem 27

Finding Intercepts In Exercises \(19-28,\) find any intercepts. \(x^{2} y-x^{2}+4 y=0\)

Short Answer

Expert verified
The x-intercept is \(x=0\) and the y-intercept is \(y=0\).

Step by step solution

01

Find the x-intercepts

To find any potential x-intercepts, plug in 0 for 'y' in the equation \(x^{2} y-x^{2}+4 y=0\).\nDoing this will give us a new equation: \(x^{2} *0 -x^{2} +4 * 0 =0\). After simplifying the equation we get \(-x^{2}=0\). Solving for x gives us \(x=0\). So, \(x=0\) is the x-intercept.
02

Find the y-intercepts

To find any potential y-intercepts, we plug in 0 for 'x'. So we get \(0^{2} y-0^{2}+4 y=0\), which simplifies to \(4y=0\). Solving for y, we find that \(y=0\) is the y-intercept.

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Key Concepts

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

x-intercept
The x-intercept is where the graph of an equation crosses the x-axis. This is a point on the graph where the y-coordinate is zero. To find the x-intercept of any given equation, you substitute y with 0 and solve for x.
In the exercise, the equation given is \(x^{2} y-x^{2}+4 y=0\). By setting \(y = 0\), the equation becomes \(-x^{2} = 0\).
This simplifies to \(x = 0\), indicating that the x-intercept is at point \((0, 0)\) on the graph.
  • Key point: For x-intercepts, always set y to 0.
  • x-intercept is where the curve crosses the x-axis.
This approach helps simplify complex equations making it easier to visualize and plot graphs.
y-intercept
The y-intercept helps us understand where a graph crosses the y-axis. At this point, the x-coordinate is zero.
Finding the y-intercept involves substituting x with 0 in the given equation and solving for y.
From the exercise, we take the equation \(x^{2} y-x^{2}+4 y=0\).Substituting \(x = 0\) gives us \(4y = 0\), which simplifies to \(y = 0\).
Thus, the y-intercept is also at the point \((0, 0)\).
  • Key point: For y-intercepts, set x to 0.
  • y-intercept indicates the point where the line meets the y-axis.
Finding y-intercepts improves our understanding of the equation's behavior along the y-axis.
equation solving
Solving equations is a fundamental skill in mathematics that allows us to find unknown values. In this context, we focused on intercepts, which often require solving the equation by substituting values strategically.
For x-intercepts, substitute y with 0 to convert the equation and solve for x. Similarly, for y-intercepts, substitute x with 0 and solve for y.
  • Ensure to follow arithmetic rules when simplifying equations.
  • Use substitution to make the process easier.
Recognizing the type of information we need (like intercepts) helps decide the right substitution and unfolds the solution step-by-step, leading us to easily find the values that meet our needs.

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Most popular questions from this chapter

Finding Composite Functions In Exercises \(63-66,\) find the composite functions \(f^{\circ} g\) and \(g \circ f\) . Find the domain of each composite function. Are the two composite functions equal? $$\begin{array}{l}{f(x)=\frac{1}{x}} \\ {g(x)=\sqrt{x+2}}\end{array}$$

Finding the Slope and \(y\) -Intercept In Exercises \(29-34\) , find the slope and the \(y\) -intercept (if possible) of the line. \(5 x+y=20\)

Choosing a Job As a salesperson, you receive a monthly salary of \(\$ 2000,\) plus a commission of 7\(\%\) of sales. You are offered a new job at \(\$ 2300\) per month, plus a commission of 5\(\%\) of sales. (a) Write linear equations for your monthly wage \(W\) in terms of your monthly sales \(s\) for your current job and your job offer. (b) Use a graphing utility to graph each equation and find the point of intersection. What does it signify? (c) You think you can sell \(\$ 20,000\) worth of a product per month. Should you change jobs? Explain.

Analyzing a Line \(A\) line is represented by the equation \(a x+b y=4\) . (a) When is the line parallel to the \(x\) -axis? (b) When is the line parallel to the \(y\) -axis? (c) Give values for \(a\) and \(b\) such that the line has a slope of \(\frac{5}{8} .\) (d) Give values for \(a\) and \(b\) such that the line is perpendicular to \(y=\frac{2}{5} x+3 .\) (e) Give values for \(a\) and \(b\) such that the line coincides with the graph of \(5 x+6 y=8 .\)

Domain Find all values of \(c\) such that the domain of $$f(x)=\frac{x+3}{x^{2}+3 c x+6}$$ is the set of all real numbers.
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