Problem 19 - Algebra A Combined Function

Question

Solve each system of linear equations by graphing. See Examples 3 through \(6 .\) \(\left\\{\begin{array}{l}y=-x-1 \\ y=2 x+5\end{array}\right.\)

Step-by-Step Solution

Answer

The solution is the point where the lines intersect: \((-2, 3)\).

Step 1: Write the equations in slope-intercept form

Both equations are already in slope-intercept form, which is \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. The equations are: 1. \( y = -x - 1 \)2. \( y = 2x + 5 \).

Step 2: Graph the first equation

For the equation \( y = -x - 1 \):- Start at the y-intercept \( (0, -1) \).- Use the slope \( m = -1 \) to find another point. From \( (0, -1) \), move down 1 unit and right 1 unit to reach \( (1, -2) \).- Draw the line through these points.

Step 3: Graph the second equation

For the equation \( y = 2x + 5 \):- Start at the y-intercept \( (0, 5) \).- Use the slope \( m = 2 \) to find another point. From \( (0, 5) \), move up 2 units and right 1 unit to reach \( (1, 7) \).- Draw the line through these points.

Step 4: Find the intersection point

Look at the graph and find where the two lines intersect. The intersection represents the solution to the system of equations.

Step 5: Verify the solution

Substitute the coordinates of the intersection point back into the original equations to ensure both equations are satisfied.

Key Concepts

System of Equations

A system of equations consists of two or more equations involving the same set of variables. In many cases, we're looking to find the set of values or points where the equations are true simultaneously. For linear systems, like the one we have in the exercise, we expect to have two linear equations and aim to find where these lines intersect on a graph. This solution often reveals the values of the variables that satisfy all equations at once.

Systems of equations can be

• Consistent, where at least one set of values satisfies all equations.
• Inconsistent, where no common solution exists.
• Dependent, where the equations represent the same line.

With graphing, you visually look to find the coordinates where both lines cross, representing the solution of the system. It is a practical way to understand solutions geometrically.

Slope-Intercept Form

The slope-intercept form is a way to write linear equations, known as \( y = mx + b \). Here, \( m \) represents the slope, and \( b \) indicates the y-intercept of the line.

Understanding these elements:

• Slope \( m \): This number shows how steep a line is. If positive, the line climbs up from left to right. If negative, it falls down.
• Y-intercept \( b \): This value is where the line cuts across the y-axis, showing the value of \( y \) at \( x = 0 \).

The first step in solving systems by graphing is expressing each equation in this form. Once in slope-intercept form, it becomes easier to manually plot the graphs of the equations, starting from the y-intercept and moving according to the slope. For example, the first equation \( y = -x - 1 \) begins at \( (0, -1) \), and the slope determines the subsequent points.

Intersection Point

The intersection point is the place on the graph where two lines meet. For a system of linear equations, this point is significant because it represents the common solution or the set of values that satisfies both equations.

When graphing two lines, visually scan for the place they cross. This is your intersection and solution.

• Draw both lines accurately.
• Identify the crossing point.
• Verify by plugging intersection coordinates back into the original equations.

In this exercise, the intersection point is where the graphs of \( y = -x - 1 \) and \( y = 2x + 5 \) meet. The coordinates of the intersection point must satisfy both linear equations. Being certain of the accuracy of your graph provides confidence in the solution you've identified.