Problem 12 - Algebra A Combined Function

Question

Solve each system of linear equations by graphing. See Examples 3 through \(6 .\) \(\left\\{\begin{array}{l}x+y=1 \\ -x+y=-3\end{array}\right.\)

Step-by-Step Solution

Answer

The solution is \((2, -1)\).

Step 1: Convert to Slope-Intercept Form

The first step is to convert each equation into the slope-intercept form, which is \( y = mx + b \). For the first equation \( x + y = 1 \), subtract \( x \) from both sides to get \( y = -x + 1 \). For the second equation \( -x + y = -3 \), add \( x \) to both sides to get \( y = x - 3 \).

Step 2: Sketch the First Line

Now, graph the first equation \( y = -x + 1 \). Start by plotting the y-intercept, which is \( (0, 1) \). Then, use the slope \( m = -1 \) (which means going down 1 unit and to the right 1 unit) to identify another point, say \( (1, 0) \). Draw the line through these points.

Step 3: Sketch the Second Line

Next, graph the second equation \( y = x - 3 \). Begin by plotting the y-intercept, which is \( (0, -3) \). Using the slope \( m = 1 \) (which means going up 1 unit and to the right 1 unit), find another point, say \( (1, -2) \). Draw the line through these points.

Step 4: Find the Intersection Point

The solution to the system is the point where the two lines intersect. From the graph, observe that the two lines intersect at the point \( (2, -1) \). This point satisfies both equations of the system.

Key Concepts

Graphing Method

The graphing method involves plotting lines on a coordinate grid to visualize the solution to a system of linear equations. This technique aids in understanding how the equations relate to each other and finding their point of intersection, which represents the solution.

To graph using this method, follow these steps:

• Convert each equation into slope-intercept form, making it easier to identify the slope and y-intercept.
• Plot the y-intercept on the graph as the first point of each line.
• Use the slope to determine additional points, ensuring accuracy in the line's direction and steepness.
• Draw the lines and visually inspect where they intersect.

This method is especially helpful for quickly checking solutions and gaining an intuitive understanding of the relationships between the equations.

Slope-Intercept Form

The slope-intercept form of a linear equation is expressed as \(y = mx + b\). It makes graphing straightforward because it directly highlights the line's slope \(m\) and its y-intercept \(b\).

- **Slope (\(m\)):** Indicates the line's steepness and direction. - A positive slope means the line rises as it moves right. - A negative slope means the line falls as it moves right.- **Y-Intercept (\(b\)):** The point where the line crosses the y-axis.By converting equations into this form:

• One can easily plot the initial y-intercept point on the graph.
• The slope can then guide the placement of the second point to draw the line accurately.

The simplicity of using slope-intercept form can demystify the process of graphing linear equations.

Intersection Point

The intersection point of two lines is the solution to a system of linear equations. It's the spot where both lines meet, implying both equations are satisfied simultaneously at this point.

To find the intersection point visually:

• Graph both equations on the same coordinate plane.
• Identify the exact point where the two lines cross.
• This point's coordinates are the solution to the system.

For systems that intersect at a single point, that point will have x and y values that solve both equations of the system. This method provides a clear visual confirmation of the solution.

Linear Equations

Linear equations represent straight lines on a graph and can be expressed in various forms, such as standard form or slope-intercept form. These equations usually involve variables with no exponents or only specific linear combinations.

Key features of linear equations include:

• Constant rate of change, represented by the slope in slope-intercept form.
• Graph results in a straight line on a coordinate plane.
• Typically takes forms like \(ax + by = c\) or \(y = mx + b\).

Understanding linear equations is crucial because they form the basis of solving systems of equations. When combined, they can represent complex relationships in real-world problems, making them an essential tool in algebra.