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Chapter 6: Problem 76

Rewrite the equation in slope-intercept form. $$x+5 y=20$$

Short Answer

Expert verified
The equation in slope-intercept form is \(y = -\frac{1}{5}x + 4\)

Step by step solution

01

Isolating Terms

The first step involves isolating \(y\) on one side. To do this, subtract \(x\) from both sides of the equation. That gives us \(5y = 20 - x\).
02

Isolate y

Next, isolate \(y\) by dividing every term of the equation by 5. Hence, the equation becomes \(y = 4 - \frac{1}{5}x\)
03

Rewrite Equation

For final step, rewrite the equation in the slope-intercept form \(y = mx + b\). So our final equation is \(y = - \frac{1}{5}x + 4\)

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Key Concepts

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

Understanding Linear Equations
Linear equations are mathematical sentences illustrating a straight-line relationship between two variables. They are foundational in algebra because they help us understand relationships where variables change at a constant rate.
One commonly used form to express linear equations is the slope-intercept form, which is beneficial for quickly identifying the slope and y-intercept of the line.
  • The general formula for slope-intercept form is \(y = mx + b\), where:
  • \(y\) and \(x\) are the variables, representing coordinates on a graph.
  • \(m\) is the slope, which indicates the steepness or tilt of the line; a positive \(m\) means an upward slope, while a negative \(m\) indicates a downward slope.
  • \(b\) is the y-intercept, the point where the line crosses the y-axis.
Understanding these components helps us graph linear equations, predict future values, and solve real-world problems involving constant rates of change.
Equation Manipulation Techniques
Equation manipulation is a crucial skill in dealing with algebraic expressions. It involves performing various operations to simplify or rearrange an equation to a desired form, such as the slope-intercept form.
To rewrite an equation like \(x + 5y = 20\) into the slope-intercept form, follow these steps:
  • Isolating terms: Begin by isolating the variable you want as the subject. In this case, isolate \(y\) by moving \(x\) to the opposite side. This is done by subtracting \(x\) from both sides, resulting in \(5y = 20 - x\).
  • Isolate your variable: After isolating terms, divide every term by the coefficient of \(y\), which is 5 in this scenario. The result is \(y = 4 - \frac{1}{5}x\).
These steps transform the equation into its desired form while maintaining the equality, allowing easier graphing and interpretation of the relationship between the variables.
Working with Algebraic Expressions
Algebraic expressions consist of variables, constants, and arithmetic operations. Understanding how to manipulate these expressions is essential in solving equations and problems efficiently.
In the context of converting equations such as \(x + 5y = 20\) to slope-intercept form, you perform specific operations involving algebraic expressions:
  • Identifying constants and coefficients: Recognize the numbers accompanying variables and stand-alone constants. In \(5y = 20 - x\), 5 is a coefficient, and 20 and \(-x\) are constants.
  • Performing operations: Use addition, subtraction, multiplication, or division to rearrange the equation. Dividing through by the coefficient of \(y\) changes the structure to give \(y\) on one side alone.
Being comfortable with these algebraic operations enables you to transform and solve equations, helping uncover the solutions to complex mathematical and real-life challenges.

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