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Chapter 12: Problem 71

Find the slope and \(y\) -intercept of the graph of the equation. $$2 x+5 y=10$$

Short Answer

Expert verified
The slope of the line is \(-\frac{2}{5}\) and the y-intercept is \(2\).

Step by step solution

01

Rewrite in slope-intercept form

The first step is to transform the given linear equation into the slope-intercept form. Beginning with \(2x+5y = 10\), subtract 2x from both sides to isolate the y-variable on one side: \[5y = -2x+10\] Then, divide each term by 5 to solve for y: \[y = -\frac{2}{5}x + 2\]
02

Identify the slope and y-intercept

Now having the equation in the form \(y = mx + b\), we can directly identify the slope \(m\) and the y-intercept \(b\). In this case, the slope \(m\) is \(-\frac{2}{5}\) and the y-intercept \(b\) is \(2\).

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Key Concepts

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

Linear Equations
A linear equation is an algebraic equation in which each term is either a constant or the product of a constant and a single variable. Linear equations can be written in various forms, but one of the most common is the slope-intercept form, expressed as (y = mx + b), where m represents the slope of the line and b represents the y-intercept.
  • It's called 'linear' because its graph is a straight line.
  • These equations can have one or more variables but the highest power of the variable must be 1.
When we work with linear equations, we often rearrange them into slope-intercept form. This is helpful because it makes it easy to graph the equation and to understand the relationship between the variables directly.
Slope of a Graph
Slope is a measure of the steepness or incline of a line. In the context of a graph, the slope is calculated as the ratio of the change in the y-coordinate (rise) to the change in the x-coordinate (run). When looking at a slope-intercept form equation, the slope is represented by m in the equation (y = mx + b).
Here are a few important points about slope:
  • A positive slope means the line inclines upwards as it moves from left to right.
  • A negative slope means the line declines as it moves from left to right.
  • If the slope is zero, the line is horizontal and describes a constant function.
  • Vertical lines have an undefined slope because their 'run' is zero.
The concept of slope is vital for understanding the relationship between two variables and plays a crucial role in graphing linear equations.
Y-Intercept
The y-intercept is the point where a line crosses the y-axis of a graph. In the slope-intercept form of a linear equation (y = mx + b), the y-intercept is represented by the constant term , which is the value of y when x equals zero.
  • It's an important characteristic of a graph because it gives you an initial value or starting point.
  • Knowing the y-intercept is especially useful when comparing two or more linear relationships.
The y-intercept can often be interpreted in real-world problems as the starting condition or inherent value of what's being evaluated, before any changes are made.
Algebraic Manipulation
Algebraic manipulation refers to the process of rearranging and simplifying algebraic expressions and equations. It involves a variety of techniques such as adding, subtracting, multiplying, dividing, factoring, and expanding. In the context of converting a standard linear equation into slope-intercept form, algebraic manipulation allows us to isolate y and express the equation in a form where the slope and y-intercept are readily identifiable.
For instance:
  1. Isolating variables: Move all terms containing the variable y to one side of the equation.
  2. Consolidating terms: Combine like terms to simplify the equation.
  3. Solving for y: Divide each term by the coefficient of y if it's not 1 to solve for it explicitly.
Mastery of algebraic manipulation is a key skill in algebra that allows us to solve equations, which in turn aids in analyzing and understanding mathematical relationships.

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Most popular questions from this chapter

Graph by using the slope and \(y\) -intercept. (GRAPH CAN'T COPY) $$y=\frac{2}{3} x$$

Graph. (GRAPH CANNOT COPY) $$x=3$$

Evaluate a function. Given \(f(x)=3 x-4,\) find \(f(4)\)

Graph by using the slope and \(y\) -intercept. (GRAPH CAN'T COPY) $$3 x-y=1$$

Determine whether the line through \(P_{1}\) and \(P_{2}\) is parallel, perpendicular, or neither parallel nor perpendicular to the line through \(Q_{1}\) and \(Q_{2}\). $$P_{1}(5,1), P_{2}(3,-2) ; Q_{1}(0,-2), Q_{2}(3,-4)$$
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