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Chapter 2: Problem 110

Do the equations \(y-2=3(x-2)\) and \(y=3 x-4\) describe the same line? Explain.

Short Answer

Expert verified
Yes, both equations describe the same line: \( y = 3x - 4 \).

Step by step solution

01

Simplify the First Equation

Start by simplifying the first equation, which is in the point-slope form, into the slope-intercept form. The point-slope form is given by \( y - y_1 = m(x - x_1) \), where \( m \) is the slope. Given equation: \( y - 2 = 3(x - 2) \)Distribute the 3: \( y - 2 = 3x - 6 \)Add 2 to both sides: \( y = 3x - 4 \).Now the equation is in slope-intercept form: \( y = 3x - 4 \).
02

Compare with the Second Equation

The second equation is already in the slope-intercept form: \( y = 3x - 4 \).Compare \( y = 3x - 4 \) from Step 1 to the second equation \( y = 3x - 4 \). They are identical.
03

Conclusion

Since both equations simplify to \( y = 3x - 4 \), they describe the same line. They have the same slope and the same y-intercept, meaning they are not just parallel, but also overlap entirely.

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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 of a linear equation is a way of writing the equation of a line so that you can easily identify its slope and y-intercept. It's written as \( y = mx + b \). Here:
  • \( m \) is the slope of the line, which tells you how steep the line is.
  • \( b \) is the y-intercept, which is where the line crosses the y-axis.
To better understand, imagine you are reading coordinates off a grid. If you have the slope-intercept form, you know exactly where the line starts and how it climbs or descends as you move along the x-axis.
When we transformed the equation from \( y - 2 = 3(x - 2) \) to \( y = 3x - 4 \), it became clear that:
  • The slope is 3, meaning for every one unit you move to the right on the x-axis, you move up 3 units on the y-axis.
  • The y-intercept is -4, showing that the line crosses the y-axis at -4.
Knowing the slope and y-intercept provides a quick snapshot of the line's positioning and direction.
Point-Slope Form
The point-slope form of a line's equation is very different from the slope-intercept form. It's expressed as \( y - y_1 = m(x - x_1) \). Here's what each symbol means:
  • \( m \) is the slope again, indicating the steepness of the line.
  • \( (x_1, y_1) \) is a specific point on the line.
This form is particularly useful when you know a point and the slope of a line, but not necessarily where it crosses the y-axis.
In the example equation \( y - 2 = 3(x - 2) \), you start with a point \( (2, 2) \) and a slope of 3. By distributing and simplifying this equation to \( y = 3x - 4 \), you then convert it to the slope-intercept form, which shows the same characteristics in a different style.
Ultimately, point-slope form focuses on giving you a head start with a reachable point and a route (slope) to follow.
Identical Equations
Identical equations are equations that represent the same line graphically. When two equations simplify to the same slope-intercept form, such as \( y = 3x - 4 \) from our exercise, they describe not just parallel lines, but precisely the same line.
  • Having the same slope means they rise and run at exactly the same angle.
  • Sharing the same y-intercept indicates they meet the y-axis at the same spot.
This sameness implies that no matter how far you extend the line in either direction, both equations trace the same pathway without deviation.
During the step-by-step process, transforming the point-slope equation into a slope-intercept one results in an identical slope and y-intercept as the other equation provided, confirming they are indeed one and the same.

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