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Chapter 2: Problem 33

Find the \(x\) - and \(y\) -intercepts of the graph of the equation. \(2 y-x y+3 x=4\)

Short Answer

Expert verified
The y-intercept is 2 and the x-intercept is \( \frac{4}{3} \).

Step by step solution

01

Finding the Y - Intercept

Set \(x = 0\) and solve for \(y\). This will result in: \[2y - 0 + 3 \cdot 0 = 4\], which simplifies to \[2y = 4\]. Solving for \(y\), you find that \(y = 2\). So, 2 is the y-intercept.
02

Finding the X - Intercept

Set \(y = 0\) and solve for \(x\). This will result in: \[2 \cdot 0 - 0 + 3x = 4\], which simplifies to \(3x = 4\). Solving for \(x\), you find that \(x = \frac{4}{3}\). So, \(\frac{4}{3}\) is the x-intercept.

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Key Concepts

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

Finding x-intercept
Finding the x-intercept of a linear equation is a useful skill in algebra. The x-intercept is the point where the line crosses the x-axis. This happens when the value of y is zero. To find the x-intercept, you substitute 0 for y in the equation and solve for x.

For example, consider the equation \(2y - xy + 3x = 4\). To find the x-intercept, set \(y = 0\) and the equation becomes \(2 \cdot 0 - 0 + 3x = 4\).

This simplifies to \(3x = 4\). Now, to solve for x, divide both sides by 3:

\[x = \frac{4}{3}\]

So the x-intercept is \(\frac{4}{3}\). This is where the line would cross the x-axis.
Finding y-intercept
Finding the y-intercept of a linear equation is similar to finding the x-intercept, but here we focus on where the line crosses the y-axis. This happens when the value of x is zero. To find the y-intercept, simply set x to 0 in the equation and solve for y.

Using the equation \(2y - xy + 3x = 4\) again, let x be 0. Substituting gives \(2y - 0 + 3 \cdot 0 = 4\).

Simplify this to obtain \(2y = 4\). To solve for y, divide both sides by 2:

\[y = 2\]

So, the y-intercept is 2. This is the point where the line crosses the y-axis.
Linear equations
Linear equations are fundamental in algebra and are represented by straight lines on a graph. The standard form of a linear equation is \(Ax + By = C\), where A, B, and C are constants.

In linear equations, both the x and y variables are raised to the first power. This means there are no x虏, y虏, or other higher power terms. The equation can also be expressed in slope-intercept form \(y = mx + b\), where m is the slope, and b is the y-intercept.

These equations are used to model relationships between two variables and can be solved using methods like substitution and elimination.
  • Linear equations have constant rates of change, represented by their slope.
  • They can be used to find intercepts that help in graphing the equation.
  • Intercepts offer practical points to start sketching the line on a coordinate plane.
Understanding how to manipulate and solve these equations is key for success in algebra.

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Most popular questions from this chapter

Describe the sequence of transformations from \(f(x)=\sqrt[3]{x}\) to \(y\). Then sketch the graph of \(y\) by hand. Verify with a graphing utility. \(y=2 \sqrt[3]{x-2}+1\)

Describe the sequence of transformations from \(f(x)=\sqrt[3]{x}\) to \(y\). Then sketch the graph of \(y\) by hand. Verify with a graphing utility. \(y=\sqrt[3]{x}-1\)

While driving at \(x\) miles per hour, you are required to stop quickly to avoid an accident. The distance the car travels (in feet) during your reaction time is given by \(R(x)=\frac{3}{4} x\). The distance the car travels (in feet) while you are braking is given by \(B(x)=\frac{1}{15} x^{2}\) Find the function that represents the total stopping distance \(T\). (Hint: \(T=R+B\).) Graph the functions \(R, B\), and \(T\) on the same set of coordinate axes for \(0 \leq x \leq 60\).

Find (a) \((f+g)(x)\), (b) \((f-g)(x)\), (c) \((f g)(x)\), and (d) \((f / g)(x)\). What is the domain of \(f / g\) ? \(f(x)=\frac{x}{x+1}, \quad g(x)=x^{3}\)

Evaluate the function for \(f(x)=2 x+1\) and \(g(x)=x^{2}-2\) \((f-g)(-2)\)
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