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Chapter 6: Problem 7

The point \(\mathrm{A}\left(x_{1}, y_{1}\right)\) lies on the line \(y=-2 x+3 .\) If the value of \(x_{1}\) is increased by 7, what is the resulting change in the value of \(y_{1}\) ?

Short Answer

Expert verified
The change in the value of \(y_1\) is -14.

Step by step solution

01

Identify Initial Equation

The line equation provided is \(y = -2x + 3\). When \(A(x_1, y_1)\) lies on this line, it satisfies this equation. Therefore, \(y_1 = -2x_1 + 3\).
02

Analyze Change in \(x\)

The problem states that the value of \(x_1\) is increased by 7. Thus, the new \(x\) value, denoted as \(x_2\), is \(x_1 + 7\).
03

Calculate New \(y\) Value

Substitute \(x_2 = x_1 + 7\) into the equation \(y = -2x + 3\). \[ y_2 = -2(x_1 + 7) + 3 \] Simplify this to find \(y_2\).
04

Simplify Expression for \(y_2\)

Expand and simplify the expression: \[ y_2 = -2x_1 - 14 + 3 = -2x_1 - 11 \]
05

Determine Change in \(y\)

The change in \(y\), denoted as \(\Delta y\), is the difference between \(y_2\) and \(y_1\). Since \(y_1 = -2x_1 + 3\), the change \(\Delta y\) is: \[ \Delta y = y_2 - y_1 = (-2x_1 - 11) - (-2x_1 + 3) \]Simplify the expression to find the change.
06

Simplify \(\Delta y\) Expression

Expand and simplify the expression for \(\Delta y\): \[ \Delta y = -2x_1 - 11 + 2x_1 - 3 = -14 \]

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Key Concepts

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

Slope
The slope of a line is a measure of its steepness and direction. In the equation of a line presented as \( y = mx + b \), the slope is represented by \( m \). Here, in the equation \( y = -2x + 3 \), the slope is \(-2\).
  • Negative slope: This means that for every unit increase in \( x \), the value of \( y \) decreases by 2 units.
  • A steeper line: A slope of \(-2\) indicates a fairly steep line, where the change in \( y \) is quite significant for a change in \( x \).
Understanding the slope is crucial because it shows how changes in \( x \) will affect \( y \). For this linear function, the consistent decrease is due to the negative slope, making it essential in predicting changes.
Equation of a Line
The equation of a line forms the basis for understanding linear functions. The standard format \( y = mx + b \) helps identify two key features: the slope \( m \) and the y-intercept \( b \). This is important in solving problems related to linear functions.
  • In the equation \( y = -2x + 3 \), the slope \( m = -2 \) affects how \( y \) changes with \( x \).
  • The y-intercept \( b = 3 \) is where the line crosses the y-axis. This means when \( x = 0 \), \( y = 3 \).
To better understand any changes or solve for specific values, finding \( y \) based on different \( x \) values, as we did in the exercise, is essential.
Change in Variables
When analyzing a linear function, changes in variables are crucial. In this exercise, increasing \( x_1 \) by 7 units directly impacts \( y \) because of the slope.
  • Calculate new \( x \): Increasing \( x_1 \) by 7 gives \( x_2 = x_1 + 7 \).
  • Impact on \( y \): Substitute \( x_2 \) into the line equation to find the new \( y \) value.
  • The change in \( y \) is calculated as \( \Delta y = y_2 - y_1 \).
In this particular example, since the slope is \(-2\), every increment in \( x \) results in a reduction of \( 2 \) times the increase in \( x \), leading to a total change in \( y \) of \(-14\). So, understanding variable changes helps in predicting outcomes and exploring more possibilities.

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