Problem 49 - Algebra A Combined Function

Question

Without graphing, decide. a. Are the graphs of the equations identical lines, parallel lines, or lines intersecting at a single point? b. How many solutions does the system have? See Examples 7 and 8 . \(\left\\{\begin{array}{l}6 y+4 x=6 \\ 3 y-3=-2 x\end{array}\right.\)

Step-by-Step Solution

Answer

The lines intersect at a single point and the system has one solution.

Step 1: Convert Equations to Slope-Intercept Form

To find the relationship between the two equations, start by converting them into slope-intercept form, \(y = mx + b\). The first equation is \(6y + 4x = 6\). Solving for \(y\), subtract \(4x\) from both sides to get \(6y = -4x + 6\). Divide each term by 6: \(y = -\frac{2}{3}x + 1\). The second equation is \(3y - 3 = -2x\). Add \(2x\) to both sides: \(3y = 2x - 3\), and then divide each term by 3: \(y = \frac{2}{3}x - 1\).

Step 2: Compare the Slopes and Y-Intercepts

For the first equation, the slope \(m_1 = -\frac{2}{3}\) and the y-intercept \(b_1 = 1\). For the second equation, the slope \(m_2 = \frac{2}{3}\) and the y-intercept \(b_2 = -1\). Since the slopes \(m_1\) and \(m_2\) are not equal, the lines are not parallel or identical. The difference in slopes indicates that the lines will intersect at a single point.

Step 3: Determine the Number of Solutions

Since the lines have different slopes, they will intersect exactly once. This means the system of equations has exactly one solution.

Key Concepts

Slope-Intercept Form

In mathematics, the slope-intercept form of a linear equation is a common way to express the equation of a line. It is written as \(y = mx + b\), where \(m\) represents the slope of the line, and \(b\) represents the y-intercept. You can think of the slope as how steep the line is and the y-intercept as the starting point of the line on the y-axis.

• Slope \(m\): Describes the tilt or incline of the line. A positive slope means the line rises as it moves to the right, whereas a negative slope means it falls.
• Y-intercept \(b\): The point where the line crosses the y-axis. It's the value of \(y\) when \(x\) is zero.

To find the slope-intercept form, you rewrite the given equation to solve for \(y\). This helps in easily identifying and comparing the slope and intercept of different lines.

Parallel Lines

Parallel lines are lines in a plane that remain the same distance apart and never intersect. For two lines to be parallel, they must have identical slopes but different y-intercepts. This means they are both going up or down at the same rate, but they start at different points on the y-axis.
For example, if you have two equations \(y = \frac{2}{3}x + 1\) and \(y = \frac{2}{3}x - 1\), these lines are parallel. They both have slopes of \(\frac{2}{3}\), meaning they rise 2 units for every 3 units they move horizontally. However, their y-intercepts are different (1 and -1, respectively). Therefore, they completely avoid intersecting with each other.

Identical Lines

Identical lines are essentially the same line presented in two different ways. This happens when two line equations have both identical slopes and identical y-intercepts. When both these parameters match, the two lines lie on top of each other, making them indistinguishable when graphed.
For example, if you were given equations like \(y = 2x + 3\) and \(2y = 4x + 6\), the second can be simplified to match the first, making them identical. This means that every point on one line also lies on the other, leading to infinite solutions in a system of equations.

Intersecting Lines

Intersecting lines meet or cross each other at a certain point. In terms of their linear equations, intersecting lines have different slopes. This slope difference guarantees that the lines are not moving parallel and, therefore, must cross somewhere along their lengths.
The point where the lines intersect is a solution to both of their equations, referred to as the system's solution. If you have equations such as \(y = -\frac{2}{3}x + 1\) and \(y = \frac{2}{3}x - 1\), as given in the exercise, their different slopes ensure that they meet at just one point. Thus, their system of equations has exactly one unique solution, which is the intersection point.