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Chapter 4: Problem 44

Determine the slope of the line. $$ 5 x+4 y-8=0 $$

Short Answer

Expert verified
The slope of the line represented by the equation \(5x + 4y - 8 = 0\) is -1.25.

Step by step solution

01

Convert the Equation into Slope-Intercept Form

In order to find the slope, first rearrange the given equation into slope-intercept form. Start with the given equation \(5x + 4y - 8 = 0\). Isolate y by subtracting 5x from both sides and then dividing all terms by 4. This results in the slope-intercept form equation: \(y = -\frac{5}{4}x + 2\).
02

Identify the Slope

From the previous step, the equation is now in the form \(y = mx + b\). In this form, the slope \(m\) is the coefficient of \(x\). The equation is \(y = -\frac{5}{4}x + 2\), hence the slope of the line is -\frac{5}{4} or -1.25.

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Key Concepts

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

Slope-Intercept Form
The slope-intercept form is a way to write a linear equation where you can easily identify the slope and the y-intercept of the line. The standard slope-intercept form is written as:\[ y = mx + b \] Here, \( m \) represents the slope of the line, while \( b \) is the y-intercept, which is the point where the line crosses the y-axis.
  • To convert any equation into slope-intercept form, you must solve for \( y \) so it's isolated on one side of the equation.
  • This form is particularly handy because it gives you a straightforward view of how steep the line is and where it cuts across the y-axis.
In the exercise, the equation \(5x + 4y - 8 = 0\) was rearranged into \(y = -\frac{5}{4}x + 2\) to reveal both the slope and the y-intercept.
Linear Equations
Linear equations are a fundamental concept in algebra that represent straight lines when graphed on a coordinate plane. They involve variables that are of the first degree, meaning they don't have exponents or powers higher than one.
  • In a linear equation like \( y = mx + b \), the graph will always be a straight line.
  • The relationship between the variables is direct; as one variable increases or decreases, the other does so in a linear manner based on the slope \( m \).
Linear equations can appear in various forms, but they all ultimately describe a straight line. By analyzing the slope and y-intercept, we can determine the direction, steepness, and initial position of those lines. The exercise demonstrates converting a linear equation to make these aspects clear.
Algebra Concepts
Algebra is a branch of mathematics dealing with symbols and the rules for manipulating those symbols. It's the foundation for understanding equations, functions, and mathematical relations.
  • One of the core concepts in algebra is solving equations, which means finding the value of the variable that makes the equation true.
  • Another important idea is manipulating equations into different forms, such as rewriting a standard form equation into slope-intercept form.
The exercise uses basic algebraic principles to transform and simplify the equation, showcasing how operations are systematically applied to make variables more manageable. These algebra skills are critical for deciphering and understanding more complex mathematical problems.

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Most popular questions from this chapter

In Exercises \(5-8\), state whether the boundary of the graph of the inequality should be dashed or solid. $$ 2 x+3 y>6 $$

In Exercises 45-48, write an equation of the line that passes through the points. Write the equation in general form. $$ (1,0.6),(2,-0.6) $$

Simplify the expression. $$ 5 x^{2}\left(x^{5}\right) $$

In Exercises \(61-64\), solve for \(y\) in terms of \(x\). $$ 4-y+x=0 $$

In Exercises \(37-40\), use the point-slope form to write an equation of the line that passes through the point and has the specified slope. Write the equation in slope-intercept form. $$ \left(0, \frac{3}{2}\right), m=\frac{2}{3} $$
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