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Chapter 7: Problem 67

Decide whether the graphs of the two equations are $$ y=4 x+3 ; 2 y-8 x=-3 $$

Short Answer

Expert verified
The two equations are not the same; while they possess the same slope, they have different y-intercepts, and hence, their graphs will not be the same.

Step by step solution

01

Simplify the Second Equation

Let's start with the second equation \(2y - 8x = -3\). To simplify this, divide the entire equation by 2 to get the y-term on one side and you'll get \(y - 4x = -1.5\).
02

Rearrange the equation

In Step 2, arrange the simplified equation in the form of \(y = mx + b\) which is standard form for a linear equation. Doing this, you get \(y = 4x + 1.5\). The slope \(m\) is 4 and the y-intercept \(b\) is 1.5.
03

Compare the Equations

After simplifying the second equation, compare it with the first one, \(y = 4x + 3\). After comparing, it is clear to see that while the slope \(m\) is the same for both, the y-intercept \(b\) is different, with the first equation having a y-intercept of 3 and the simplified second one having a y-intercept of 1.5.

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Key Concepts

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

Graphing
Graphing linear equations is a fundamental skill in mathematics that allows us to visually interpret relationships between variables. Each linear equation is represented as a straight line on a coordinate plane.
The equation takes the form \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. To graph an equation:
  • Identify the y-intercept on the y-axis where \(x = 0\).
  • Use the slope, which is the ratio of the change in \(y\) over the change in \(x\) (rise over run), to find another point on the line.
  • Draw a line through these points extending in both directions.
Graphing helps us analyze two equations by comparing their slopes and y-intercepts. This comparison reveals how the lines lie relative to each other, whether they are parallel, perpendicular, or intersecting.
Slope
The slope of a line in a linear equation indicates its steepness and direction. Mathematically, it is represented by \(m\) in the equation \(y = mx + b\).
The slope is calculated as the ratio of the vertical change to the horizontal change between two points, often described as "rise over run." Thus, if the slope is 4, for example, it means for every unit increase in \(x\), \(y\) increases by 4 units.
  • A positive slope means the line rises as you move from left to right.
  • A negative slope means the line falls as you move from left to right.
  • A zero slope results in a horizontal line.
  • An undefined slope indicates a vertical line.
In comparing two lines, matching slopes imply parallel lines, whereas differing slopes suggest the lines will intersect.
Y-Intercept
The y-intercept of a line is the point where the line crosses the y-axis. In the equation \(y = mx + b\), \(b\) is the y-intercept. It tells us the value of \(y\) when \(x\) is zero.
Understanding the y-intercept is crucial in identifying where the line starts on the y-axis. Different y-intercepts between two lines suggest they are not the same line and will not coincide unless their entire equations are identical.
  • The y-intercept is a starting point for graphing a line.
  • Changes in the y-intercept will shift the line up or down vertically without altering its slope.
Examining the y-intercepts of two equations helps determine their relative positioning vertically on the graph. If two lines have the same slope but different y-intercepts, they are parallel and will never meet.

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