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Chapter 3: Problem 98

Find the slope of a line parallel to the line \(-3 x+4 y=10\)

Short Answer

Expert verified
The slope of a line parallel to \(-3x + 4y = 10\) is \(\frac{3}{4}\).

Step by step solution

01

Identify the Given Equation's Form

The given equation is \(-3x + 4y = 10\). This is in standard form \(Ax + By = C\), where \(A = -3\) and \(B = 4\). Our aim is to find the slope of this line first.
02

Convert to Slope-Intercept Form

To find the slope, convert the equation to slope-intercept form \(y = mx + b\) where \(m\) is the slope. Start by isolating \(y\) in \(-3x + 4y = 10\).
03

Solve for y

Add \(3x\) to both sides: \(4y = 3x + 10\). Now, divide each term by \(4\) to solve for \(y\): \(y = \frac{3}{4}x + \frac{10}{4}\). The slope \(m\) is \(\frac{3}{4}\).
04

Identify the Slope for Parallel Line

Lines that are parallel share the same slope. Therefore, the slope of a line parallel to the given equation is \(\frac{3}{4}\).

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Key Concepts

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

Parallel Lines
Parallel lines play a crucial role in geometry and are defined as lines that never meet, regardless of how far they are extended. These lines are always the same distance apart. This is why their key attribute is having equal slopes.
When we talk about slopes, we refer to the steepness or incline of a line. For parallel lines, the rule is simple: if you know the slope of one line, the slope of any line parallel to it is exactly the same.
In our exercise, we found the slope of one line to be \(\frac{3}{4}\), which means that any line parallel to it will also have a slope of \(\frac{3}{4}\).
  • Always remember: parallel lines = same slope
  • Visualize parallel lines as train tracks - they never cross!
  • Identifying parallel lines helps in understanding geometric alignment and is also key in architectural designs.
Slope-Intercept Form
The slope-intercept form of a linear equation is perhaps the most intuitive and widely used form. It is written as \( y = mx + b \). Here, \(m\) represents the slope of the line, and \(b\) is the y-intercept, which is where the line crosses the y-axis. This formula allows us to quickly identify how steep the line is and its vertical starting point.
For our exercise, converting from standard form to slope-intercept form made it easier to identify the slope of the line.
  • Converting to slope-intercept form provides a clear visual of a line鈥檚 angle.
  • The y-intercept \(b\) represents the value of \(y\) when \(x = 0\).
  • This form is especially useful for graphing, as you can start at \(b\) on the y-axis and move up \(m\) units for every unit along the x-axis.
Standard Form
The standard form of a line's equation, written as \(Ax + By = C\), offers a format that emphasizes the relationship between the x and y variables.
This form is especially handy in mathematical situations where clear separation of terms is needed, such as finding intersections between lines.
In our exercise, the given equation was already in standard form, making it straightforward to identify the coefficients needed for conversion into another format.
  • Standard form helps in organizing terms, crucial for some algebraic operations.
  • Equations in standard form make it easy to spot if two lines might be parallel or perpendicular.
  • Being able to manipulate between forms enhances flexibility and problem-solving skills in algebra.

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$$ \text { Graph: } f(x)=\left\\{\begin{array}{rll} {-\frac{1}{3} x} & {\text { if }} & {x \leq 0} \\ {x+2} & {\text { if }} & {04} \end{array}\right. $$
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