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Chapter 5: Problem 38

Solve the given problems. Find the intercepts of the line \(\frac{2 x}{3}-\frac{5 y}{2}=1\).

Short Answer

Expert verified
The x-intercept is \( x = \frac{3}{2} \) and the y-intercept is \( y = -\frac{2}{5} \).

Step by step solution

01

Find the x-intercept

The x-intercept occurs when the y-value is zero. Start by substituting 0 for \( y \) in the equation \( \frac{2x}{3} - \frac{5y}{2} = 1 \):\[\frac{2x}{3} - \frac{5(0)}{2} = 1 \rightarrow \frac{2x}{3} = 1.\]Next, solve for \( x \) by multiplying both sides by 3:\[2x = 3.\]Finally, divide both sides by 2 to get the x-intercept:\[x = \frac{3}{2}.\]
02

Find the y-intercept

To find the y-intercept, substitute 0 for \( x \) in the equation:\[\frac{2(0)}{3} - \frac{5y}{2} = 1 \rightarrow -\frac{5y}{2} = 1.\]Solve for \( y \) by multiplying both sides by -2:\[5y = -2.\]Next, divide both sides by 5:\[y = -\frac{2}{5}.\]

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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 a point on a graph where the line crosses the x-axis. This is fundamental in understanding how a line behaves in a coordinate plane. To find the x-intercept of a given line, set the y-value to zero in the equation. For example, in the linear equation \( \frac{2x}{3} - \frac{5y}{2} = 1 \), you substitute \( y = 0 \), simplifying to \( \frac{2x}{3} = 1 \). Solve this by performing operations to isolate \( x \):
  • Multiply both sides by 3: \( 2x = 3 \).
  • Divide by 2 to solve for \( x \): \( x = \frac{3}{2} \).
Thus, the x-intercept is \( x = \frac{3}{2} \), or \( (\frac{3}{2}, 0) \). Understanding this means recognizing how crucial substituting zero can simplify finding intercepts in any given linear equation.
Y-Intercept
The y-intercept is where a line intersects the y-axis, indicating what happens when \( x \) is zero. Finding the y-intercept follows a similar process to finding the x-intercept but with roles reversed. Set \( x = 0 \) in the equation. Using our previous example, substitute to get \( \frac{2(0)}{3} - \frac{5y}{2} = 1 \), which simplifies to \( -\frac{5y}{2} = 1 \). Isolating \( y \) involves solving:
  • Multiply both sides by -2: \( 5y = -2 \).
  • Divide by 5: \( y = -\frac{2}{5} \).
So, the y-intercept is \( y = -\frac{2}{5} \), or \( (0, -\frac{2}{5}) \). Recognizing intercepts allows for a more comprehensive visual understanding of linear relationships.
Linear Equations
Linear equations represent relationships between two variables, typically \( x \) and \( y \). These equations have constant rates of change, resulting in straight-line graphs. A common form is \( ax + by = c \), where \( a \), \( b \), and \( c \) are constants. In our example, the equation \( \frac{2x}{3} - \frac{5y}{2} = 1 \) is linear. Such equations are characterized by:
  • Having at most one x-intercept and y-intercept unless the line coincides with one of the axes.
  • Offering a way to understand real-world relationships through slope and intercepts.
The simplicity of linear equations facilitates learning about more advanced algebraic concepts. They provide an accessible introduction to graphing and analyzing the interaction of variables.

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Most popular questions from this chapter

Set up appropriate systems of two linear equations and solve the systems algebraically. All data are accurate to at least two significant digits. In a test of a heat-seeking rocket, a first rocket is launched at \(2000 \mathrm{ft} / \mathrm{s},\) and the heat-seeking rocket is launched along the same flight path 12 s later at a speed of \(3200 \mathrm{ft} / \mathrm{s}\). Find the times \(t_{1}\) and \(t_{2}\) of flight of the rockets until the heat- seeking rocket destroys the first rocket.

Solve the given systems of equations by the method of elimination by addition or subtraction. $$\begin{aligned} &12 t+9 y=14\\\ &6 t=7 y-16 \end{aligned}$$

Solve the indicated or given systems of equations by an appropriate algebraic method. A \(6.0 \%\) solution and a \(15.0 \%\) solution of a drug are added to \(200 \mathrm{mL}\) of a \(20.0 \%\) solution to make \(1200 \mathrm{mL}\) of a \(12.0 \%\) solution for a proper dosage. The equations relating the number of milliliters of the added solutions are \(x+y+200=1200\) \(0.060 x+0.150 y+0.200(200)=0.120(1200)\) Find \(x\) and \(y\) (to three significant digits).

Evaluate the given third-order determinants. $$\left|\begin{array}{rrr} 25 & 18 & -50 \\ -30 & 40 & -12 \\ -20 & 55 & -22 \end{array}\right|$$

Set up appropriate systems of two linear equations and solve the systems algebraically. All data are accurate to at least two significant digits. What conclusion can you draw from a sales report that states that "sales this month were \(\$ 8000\) more than lass month, which means that total sales for both months are \(\$ 4000\) more than twice the sales last month"?
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