Question

practice
examples 1 and 2
use the graph to determine the number of solutions the system
has. then state whether the system of equations is consistent or
inconsistent and if it is independent or dependent.

1. $y = x - 1$

$y = -x + 1$

1. $x - y = -4$

$y = x + 4$

1. $y = x + 4$

$2x - 2y = 2$

1. $y = 2x - 3$

$2x - 2y = 2$
examples 3 and 4
determine the number of solutions the system has. then state whether the system
of equations is consistent or inconsistent and if it is independent or dependent.

1. $y = \frac{1}{2}x$

$y = x + 2$

1. $4x - 6y = 12$

$-2x + 3y = -6$

1. $8x - 4y = 16$

$-5x - 5y = 5$

1. $2x + 3y = 10$

$4x + 6y = 12$

1. $y = -\frac{3}{2}x + 5$

$y = -\frac{2}{3}x + 5$

1. $y = x - 3$

$y = -4x + 3$
examples 5 and 6
graph each system and determine the number of solutions it has. if it has one
solution, determine its coordinates.

1. $y = -3$

$y = x - 3$

1. $y = 4x + 2$

$y = -2x - 4$

1. $y = x - 6$

$y = x + 2$

1. $x + y = 4$

$3x + 3y = 12$

1. $x - y = -2$

$-x + y = 2$

1. $2x + 3y = 12$

$2x - y = 4$
lesson 7 - 1 · graphing systems

Explanation:

Let's solve problem 1: \(

$$\begin{cases} y = x - 1 \\ y = -x + 1 \end{cases}$$

\)

Step 1: Analyze the slopes and y-intercepts

The first equation \( y = x - 1 \) is in slope - intercept form \( y=mx + b \), where the slope \( m_1=1 \) and the y - intercept \( b_1=- 1 \). The second equation \( y=-x + 1 \) has a slope \( m_2=-1 \) and a y - intercept \( b_2 = 1 \). Since \( m_1
eq m_2 \), the two lines are not parallel and will intersect at exactly one point.

Step 2: Determine consistency and dependence

A system of linear equations is consistent if it has at least one solution. Since the two lines intersect (one solution), the system is consistent. A system is independent if the two equations represent two distinct lines (not coinciding). Since the two lines have different slopes (and different y - intercepts), they are distinct lines, so the system is independent.

Step 1: Rewrite the first equation

Rewrite \( x - y=-4 \) in slope - intercept form. Solve for \( y \): \( y=x + 4 \).

Step 2: Compare the two equations

The second equation is also \( y=x + 4 \). This means that the two equations represent the same line. So, the two lines are coinciding.

Step 3: Determine the number of solutions, consistency and dependence

If two lines are coinciding, they have infinitely many solutions. A system with infinitely many solutions is consistent (because it has solutions) and dependent (because the two equations are not independent, they represent the same line).

Step 1: Rewrite the second equation in slope - intercept form

Start with \( 2x-2y = 2 \). Subtract \( 2x \) from both sides: \( - 2y=-2x + 2 \). Divide both sides by - 2: \( y=x - 1 \).

Step 2: Analyze the slopes

The first equation \( y=x + 4 \) has a slope \( m_1 = 1 \) and the second equation \( y=x - 1 \) has a slope \( m_2=1 \). The y - intercepts are \( b_1 = 4 \) and \( b_2=-1 \). Since the slopes are equal (\( m_1=m_2 \)) and the y - intercepts are different (\( b_1
eq b_2 \)), the lines are parallel.

Step 3: Determine the number of solutions, consistency and dependence

Parallel lines never intersect, so the system has no solutions. A system with no solutions is inconsistent.

Answer:

Number of solutions: 1. The system is consistent and independent.

Let's solve problem 2: \(

$$\begin{cases} x - y=-4 \\ y=x + 4 \end{cases}$$

\)