Problem 40 - 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 . \(\left\\{\begin{array}{l}3 x+y=1 \\ 3 x+2 y=6\end{array}\right.\)

Step-by-Step Solution

Answer

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

Step 1: Identify the Lines

Let's write down the equations for clarity: Equation (1): \(3x + y = 1\)Equation (2): \(3x + 2y = 6\).

Step 2: Convert to Slope-Intercept Form

Convert both equations into the slope-intercept form (\(y = mx + b\)).For Equation (1):\(3x + y = 1\)\(y = -3x + 1\)For Equation (2):\(3x + 2y = 6\)\(2y = -3x + 6\)\(y = -\frac{3}{2}x + 3\).

Step 3: Compare the Slopes

In Equation (1), the slope is \(-3\).In Equation (2), the slope is \(-\frac{3}{2}\).Since the slopes are different, the lines are not parallel.

Step 4: Determine the Number of Solutions

Since the lines have different slopes, they will intersect at a single point. Thus, the system has exactly one solution.

Key Concepts

Slope-Intercept Form

The slope-intercept form of a linear equation is one of the most intuitive ways to understand the behavior of a line on a graph. The general formula is given by \( y = mx + b \), where \( m \) represents the slope of the line, and \( b \) is the y-intercept, the point where the line crosses the y-axis.
By using the slope-intercept form, you can quickly determine the steepness and direction of a line.
- **Slope (\( m \))**: This value tells us how steep the line is. A larger absolute value of \( m \) means a steeper line, while an \( m \) of zero means the line is perfectly horizontal.- **Y-Intercept (\( b \))**: This shows where the line hits the y-axis. If \( b = 0 \), the line passes through the origin (0,0).Converting equations to this form makes it easier to compare lines, find intersections, and solve systems of equations graphically. As was done in the exercise, this form allowed us to directly compare the slopes of the given equations.

Systems of Equations

A system of equations involves solving for variables where two or more equations set constraints on those variables. The main goal is to find a solution that satisfies all the equations simultaneously.
When graphed, the system of equations can have different outcomes:

• One Solution: The lines intersect at a single point, indicating that there is exactly one set of solutions. This is what was identified in the exercise for the system \( \left\{ \begin{array}{l} 3x+y=1 \ 3x+2y=6 \end{array} \right. \).
• No Solution: The lines are parallel and distinct, meaning they extend indefinitely without ever crossing. This implies there is no common solution.
• Infinite Solutions: The lines are identical, which means they lie on top of one another on the graph. Every point on the line is a solution.

Identifying the number of solutions helps us understand the nature and relationship of the involved equations.

Parallel and Intersecting Lines

Knowing whether lines are parallel or intersecting is crucial to understanding how systems of equations behave.
- **Parallel Lines:** Two lines are parallel if they have the same slope but different y-intercepts. Parallel lines never meet, no matter how far they are extended. In terms of systems of equations, this results in no solutions.- **Intersecting Lines:** Lines that intersect have different slopes. This results in one intersection point where the solution to the system lies.In the exercise, calculating different slopes from the equations \( y = -3x + 1 \) and \( y = -\frac{3}{2}x + 3 \) confirmed that the lines intersect at a single point, because their slopes are not equal.
Identifying whether lines are parallel or intersecting assists in determining if a unique, no, or infinite solutions exist for the system.