Problem 4 - Algebra 1: Concepts and Skills

Question

Use the linear system below. $$ \begin{array}{l} {-x+y=-2} \\ {2 x+y=10} \end{array} $$ Graph both equations in the same coordinate plane.

Step-by-Step Solution

Answer

The graph of the two linear equations intersects at point (4, 2), which is the solution of the given system of equations.

Step 1: Rearrange each equation for y

To graph an equation, it's often easier when it's in slope-intercept form, \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. So, first, rearrange both equations for \( y \).\nFor the first equation, adding \( x \) to both sides gets \( y = x - 2 \). For the second equation, subtract \( 2x \) from both sides to get \( y = -2x + 10 \).

Step 2: Identify Slope and Y-Intercept

From the rearranged form, we can identify the slope \( m \) and y-intercept \( b \) of the lines. From \( y = x - 2 \), the slope \( m = 1 \) and y-intercept \( b = -2 \). From \( y = -2x + 10 \), we get slope \( m = -2 \) and y-intercept \( b = 10 \).

Step 3: Plot Y-Intercepts

The y-intercept (\( b \)) is the point at which the line crosses the y-axis. So, mark points at \( y = -2 \) for equation 1 and \( y = 10 \) for equation 2.

Step 4: Use the Slope to Draw the Lines

The slope (\( m \)) is the rise over the run, or how much \( y \) increases for each increase in \( x \). From the y-intercept for equation 1, move 1 unit up and 1 right (\( m = 1 = 1/1 \)) to mark the next point, and repeat to draw the line. For equation 2, from the y-intercept, move 2 units down and 1 unit right (\( m = -2 = -2/1 \)) to mark the next point and repeat to draw the line.

Step 5: Identify Intersection

The point at which both lines intersect is their solution. On the graph, both lines intersect at point (4, 2), which is the solution of the system of equations.

Key Concepts

Slope-Intercept Form

The slope-intercept form is a straightforward and popular way to express a linear equation. It is represented as \( y = mx + b \). Here, \( m \) is the slope of the line, which tells us how steep the line is, and \( b \) is the y-intercept, indicating where the line intersects the y-axis. This form simplifies the process of graphing equations. Some reasons why we prefer slope-intercept form include:

• Easy identification of the slope and y-intercept without further rearrangement.
• Simple visualization of how the line will look on a graph.

Consider the equation \( y = x - 2 \). Here, the slope \( m = 1 \) implies the line rises 1 unit for every 1 unit it moves to the right. The y-intercept \( b = -2 \) tells us the line crosses the y-axis at -2, making graphing more intuitive.

Graphing Equations

Graphing linear equations enables us to visualize the relationship between two variables. To graph a linear equation from its slope-intercept form, follow these steps:1. **Identify the y-intercept**, \( b \), which is where the line will intersect the y-axis. This is our starting point on the graph.2. **Use the slope** \( m \) to find another point on the line. The slope is expressed as "rise over run."For instance, consider the equation \( y = -2x + 10 \):

• The y-intercept is 10, so begin plotting at (0, 10) on the graph.
• The slope is -2, which means for every 1 unit moved right, the line goes down 2 units. This helps in plotting additional points, such as moving from (0, 10) to (1, 8). Continue this pattern to complete the graph of the line.

Graphing helps visually confirm whether two lines intersect and where that intersection occurs.

Intersection of Lines

Intersection points in a graph of linear equations are valuable as they represent the solution to a system of equations. Simply put, the intersection is the point where two lines on a graph meet or cross. To identify where lines intersect:

• Graph each equation on the same plane using the slope-intercept form.
• The intersection point can be found visually where both lines meet.

In our example, the equations \( y = x - 2 \) and \( y = -2x + 10 \), when graphed, intersect at the point (4, 2). This means that both equations have the values of \( x = 4 \) and \( y = 2 \) in common, and thus (4, 2) is the solution of this linear system.Understanding the intersection gives insight into how two relationships connect within a graph, making it a critical technique in solving linear systems of equations.