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Chapter 1: Problem 11

Find the \(x\) - and \(y\) -intercepts of the graph of the equation. $$ 2 x-y-3=0 $$

Short Answer

Expert verified
The x-intercept is \(x=1.5\) and the y-intercept is \(y=-3\).

Step by step solution

01

Find the x-intercept

Set \(y = 0\) in the equation and solve for \(x\). So, the equation becomes \(2x - 0 - 3 = 0\). Solving this equation gives \(x=1.5\).
02

Find the y-intercept

Set \(x = 0\) in the equation and solve for \(y\). So, the equation becomes \(2*0 - y - 3 = 0\). Solving this equation gives \(y=-3\).

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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 of a graph is where it crosses the x-axis. To find the x-intercept, we set the equation's y-value to zero and solve for x. Why is that? Because on the x-axis, the value of y is always zero. Let's apply this to our equation, \(2x - y - 3 = 0\). By substituting y with 0, we get \(2x - 3 = 0\), which simplifies to \(x = 1.5\). This means our graph crosses the x-axis at \((1.5, 0)\).

Understanding x-intercepts is crucial for sketching graphs and analyzing their behavior. Remember, the x-intercept is not always a single point; some graphs, like horizontal lines, can have no x-intercept or infinitely many!
Y-Intercept
In contrast to the x-intercept, the y-intercept is the point where the graph intersects the y-axis. To find it, you simply let x be zero in the equation, because anywhere on the y-axis, the x-value is zero. Returning to our example, if we take \(2x - y - 3 = 0\) and make x zero, the equation becomes \(-y - 3 = 0\), which solves to \(y = -3\). Therefore, our y-intercept is \((0, -3)\).

The y-intercept gives you a starting point for graphing linear equations and is also a key concept in understanding the function's initial value鈥攖he starting value when the input (x) is zero.
Linear Equations
Linear equations, like our example \(2x - y - 3 = 0\), form straight lines when graphed. They are beautifully simple, with a general form of \(Ax + By = C\), where A, B, and C are constants. Here's a quick breakdown of their properties:
  • They have a constant rate of change (slope).
  • They generate straight lines on a graph.
  • Any line on a graph can be represented through a linear equation, and vice versa.
Linear equations are foundational in algebra, impacting various disciplines, including economics, engineering, and the sciences, because they model relationships where each variable has a consistent effect on the other.
Solving Equations
Solving equations is like a treasure hunt; you're seeking the value(s) that make the equation true. The process involves a series of algebraic steps to isolate the variable. For our example equation \(2x - y - 3 = 0\), solving for x when y is zero (and vice versa) gives us the intercepts. The key steps in solving equations include:
  • Combining like terms.
  • Using inverse operations to isolate the variable.
  • Checking your solution by plugging back into the original equation.
It's vital to understand solving equations for analyzing real-world problems where you have knowns and unknowns, especially in fields like science, finance, and statistics.

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Most popular questions from this chapter

You deposit \(\$ 2000\) in an account that is compounded quarterly at an annual rate of \(r\) (in decimal form). The balance \(A\) after 10 years is \(A=2000\left(1+\frac{r}{4}\right)^{40}\). Does the limit of \(A\) exist as the interest rate approaches \(6 \% ?\) If so, what is the limit?

Find the limit of (a) \(f(x)+g(x),\) (b) \(f(x) g(x),\) and \((c) f(x) / g(x),\) as \(x\) approaches \(c .\) \(\lim _{x \rightarrow c} f(x)=\frac{3}{2}\) \(\lim _{x \rightarrow c} g(x)=\frac{1}{2}\)

Use a graphing utility to graph the function and estimate the limit. Use a table to reinforce your conclusion. Then find the limit by analytic methods. \(\lim _{x \rightarrow 1^{+}} \frac{5}{1-x}\)

Find the limit (if it exists). \(\lim _{\Delta t \rightarrow 0} \frac{(t+\Delta t)^{2}-4(t+\Delta t)+2-\left(t^{2}-4 t+2\right)}{\Delta t}\)

Find the limit (if it exists). \(\lim _{\Delta x \rightarrow 0} \frac{2(x+\Delta x)-2 x}{\Delta x}\)
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