Question

consider the system of equations shown in the graph.
\

$$\begin{cases} y = x - 1 \\\\ -2x + y = 1 \\end{cases}$$

a. estimate the coordinates of the point of intersection of the lines.
b. determine whether your estimate from part (a) is the solution to the system.

Explanation:

Part (a)

Step1: Analyze the two lines

We have two linear equations: \( y = x - 1 \) and \( - 2x + y = 1 \). To find the intersection point, we can also solve the system algebraically and then check the graph, or estimate from the graph. From the graph, we look for the point where the two lines cross each other. By observing the grid, we can see that the lines intersect at a point where \( x=-2 \) and \( y=-3 \) (we can check this by substituting into the equations later, but for estimation from the graph, we look at the coordinates on the grid).

Step2: Confirm the intersection visually

Looking at the graph, the two lines cross at a point. Let's check the x - coordinate and y - coordinate. For the line \( y=x - 1 \), when \( x=-2 \), \( y=-2 - 1=-3 \). For the line \( - 2x + y = 1 \), when \( x = - 2 \), we have \( -2\times(-2)+y=1\), \( 4 + y=1\), \( y=1 - 4=-3 \). So the estimated intersection point from the graph is \((-2,-3)\).

Step1: Substitute the estimated point into the first equation

We substitute \( x=-2 \) and \( y = - 3 \) into the equation \( y=x - 1 \).
Left - hand side (LHS): \( y=-3 \)
Right - hand side (RHS): \( x - 1=-2 - 1=-3 \)
Since \( LHS = RHS \), the point \((-2,-3)\) satisfies the first equation.

Step2: Substitute the estimated point into the second equation

We substitute \( x=-2 \) and \( y=-3 \) into the equation \( - 2x + y = 1 \).
Left - hand side (LHS): \( -2\times(-2)+(-3)=4 - 3 = 1 \)
Right - hand side (RHS): \( 1 \)
Since \( LHS=RHS \), the point \((-2,-3)\) satisfies the second equation.

Answer:

The coordinates of the point of intersection are \((-2,-3)\)

Part (b)