Problem 44 - 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 . \(\left\\{\begin{array}{l}3 x-y=2 \\ \frac{1}{3} y=-2+3 x\end{array}\right.\)

Step-by-Step Solution

Answer

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

Step 1: Rewriting the second equation

We first rewrite the second equation to match the standard form \( Ax + By = C \). Given \( \frac{1}{3} y = -2 + 3x \), multiply everything by 3 to clear the fraction: \( y = -6 + 9x \).

Step 2: Express in standard form

Rearrange \( y = -6 + 9x \) to the standard form by bringing \( 9x \) to the left side: \(-9x + y = -6\). Now both equations are: \( 3x - y = 2 \) and \(-9x + y = -6\).

Step 3: Compare the coefficients

For two equations \( A_1x + B_1y = C_1 \) and \( A_2x + B_2y = C_2 \), they represent parallel lines if \( \frac{A_1}{A_2} = \frac{B_1}{B_2} eq \frac{C_1}{C_2} \). So, check \( \frac{3}{-9} = \frac{-1}{1} \) which simplifies to \( \frac{-1}{3} = -1 \). The inequality in ratios confirms the lines are neither identical nor parallel.

Step 4: Determine the intersection condition

Since the coefficients do not satisfy conditions for parallel or identical lines, the lines intersect at a single point. Therefore, the system has one solution.

Key Concepts

Linear Equations

Linear equations describe relationships between two variables, typically represented as "x" and "y." These equations are graphed as straight lines on a coordinate plane. The general form of a linear equation is \( y = mx + b \), where:

• \( m \) is the slope, showing the change in "y" for a unit change in "x".
• \( b \) is the y-intercept, where the line crosses the y-axis.

To solve a system of linear equations, we're finding the set of values that satisfy both equations simultaneously.

In the given problem, the system consists of two equations. By comparing them, we can determine how their lines behave on the graph. This involves determining if they are parallel, identical, or intersect at some point.

Intersecting Lines

When two linear equations intersect, their graphs cross each other at one point on the coordinate plane. This point represents the one solution that satisfies both equations.

For two lines to intersect, they must have different slopes. If the slopes are the same but the intercepts are different, the lines are parallel and will never touch.In the step-by-step solution, the system was checked for conditions of parallelism and identical lines. It was determined that the lines intersect at a single point, as the ratio of coefficients for \( A \) and \( B \) between the two equations was not equal, indicating different slopes. Thus, the system has exactly one solution, corresponding to the intersection point.

Standard Form of Equations

The standard form of a linear equation is expressed as \( Ax + By = C \). This format is particularly useful for analyzing and comparing multiple equations, as it highlights the coefficients directly.Transforming an equation into standard form involves rearranging its terms to feature "x" and "y" on one side of the equation and the constant on the other. This process allows us to easily manipulate equations to identify relationships such as parallelism or intersection.

In our exercise, the second equation was reworked into standard form as \( -9x + y = -6 \). Aligning both equations in this form made it easier to compare their coefficients, leading to the conclusion about how the lines behave on the graph. By making both equations match the format, the task of solving the system by comparing coefficients becomes systematic.