Problem 67 - Algebra for College Students

Question

Suppose that a line has a slope of \(\frac{2}{3}\) and contains the point \((4,7)\). Are the points \((7,9)\) and \((1,3)\) also on the line? Explain your answer.

Step-by-Step Solution

Answer

(7, 9) is on the line; (1, 3) is not on the line.

Step 1: Understand Slope-Intercept Form

The slope-intercept form of a line is given by the equation \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. Here, the slope \( m = \frac{2}{3} \).

Step 2: Use Known Point to Find y-Intercept

We can find the y-intercept \( b \) by substituting the slope \( \frac{2}{3} \) and the point \((4, 7)\) into the equation \( y = mx + b \).\( 7 = \frac{2}{3}(4) + b \).\( 7 = \frac{8}{3} + b \).Subtract \( \frac{8}{3} \) from both sides: \( b = 7 - \frac{8}{3} \).Convert \( 7 \) to \( \frac{21}{3} \) to subtract: \( b = \frac{21}{3} - \frac{8}{3} = \frac{13}{3} \).The equation of the line is \( y = \frac{2}{3}x + \frac{13}{3} \).

Step 3: Verify Point (7,9) on the Line

Substitute \( x = 7 \) into the line equation \( y = \frac{2}{3}x + \frac{13}{3} \):\( y = \frac{2}{3}(7) + \frac{13}{3} \).\( y = \frac{14}{3} + \frac{13}{3} \).\( y = \frac{27}{3} = 9 \).So, the point \((7, 9)\) is on the line.

Step 4: Verify Point (1,3) on the Line

Substitute \( x = 1 \) into the line equation \( y = \frac{2}{3}x + \frac{13}{3} \):\( y = \frac{2}{3}(1) + \frac{13}{3} \).\( y = \frac{2}{3} + \frac{13}{3} \).\( y = \frac{15}{3} = 5 \).Since \( 5 eq 3 \), the point \((1, 3)\) is not on the line.

Key Concepts

Linear Equations

A **linear equation** forms the backbone of algebra and coordinate geometry. It represents a straight line in the coordinate plane. The most common form is the standard slope-intercept form, which is expressed as \( y = mx + b \). Here, \( y \) is the dependent variable and changes based on the value of \( x \), the independent variable. The term \( m \) is known as the \( \(\textit{slope}\) \) of the line. It indicates the rise over run, or how steep the line is. The slope is calculated by \( \frac{\text{change in } y}{\text{change in } x} \). On the other hand, \( b \), the \( \(\textit{y-intercept}\) \), is the value where the line crosses the y-axis. Linear equations simplify the process of predicting one quantity from another and generally have no powers or complicated terms, making them easy to work with for both learning and application.
In the context of the exercise, our linear equation was formed using the given point \((4, 7)\) and the slope \(\frac{2}{3}\) to determine \( b \), resulting in the equation \( y = \frac{2}{3}x + \frac{13}{3} \). This equation is then used to check if other points lie on the same line.

Coordinate Geometry

**Coordinate Geometry** involves the study of geometric figures through a coordinate plane, a system that originated in the work of René Descartes. This field combines algebra with geometry to gain insights into shapes, lines, and figures.
In a coordinate system, every point is given as an ordered pair \((x, y)\). The \( x \)-coordinate specifies the horizontal position, while the \( y \)-coordinate specifies the vertical position. By using simple algebraic equations, we can describe the positions and relations of lines and shapes in this plane.
In our exercise, coordinate geometry allows us to assess whether given points, such as \((7, 9)\) and \((1, 3)\), fall on a line defined by the equation \( y = \frac{2}{3}x + \frac{13}{3} \). We substitute these specific \( x \) values into the equation to see if we get the correct \( y \) values for those points. This process is essential for validating positions and relationships of geometric figures within coordinate spaces.

Equation of a Line

The **Equation of a Line** is crucial for understanding how lines are depicted mathematically. It provides clear insight into the direction and position of a line on the Cartesian plane. Derived often in the form \( y = mx + b \), it gives us two key pieces of information: the slope \( m \) and y-intercept \( b \).
The slope \( m \) dictates how the line inclines or declines with each step to the right. A positive slope means the line rises from left to right, while a negative slope means it falls. The steeper the slope, the more vertical the line appears. Meanwhile, the y-intercept \( b \) identifies where the line meets the y-axis, indicating its starting point when \( x = 0 \).
In the exercise, we derived the equation \( y = \frac{2}{3}x + \frac{13}{3} \) to portray our line graphically. We checked the points \((7, 9)\) and \((1, 3)\) for collinearity. Based on substitutions, only \((7, 9)\) lies on this line because substituting \( x = 7 \) resulted in a true statement when \( y \) was calculated. Thus, the equation confirms the relationship of points along a line.