Let's solve the system of linear equations graphically. The equations provided are:

1. $y = -x - 4$
2. $x - 2y = -4$

Step 1: Rewrite the equations

Equation 1: $y = -x - 4$

This equation is already in slope-intercept form ($y = mx + b$), where:

• Slope ($m$) = $-1$
• Y-intercept ($b$) = $-4$

We can plot this line by starting at $(0, -4)$ (the y-intercept) and using the slope to move:

• Down 1 unit and right 1 unit (slope = $-1$).

Equation 2: $x - 2y = -4$

We rewrite this into slope-intercept form ($y = mx + b$): $x - 2y = -4$ $-2y = -x - 4$ $y = \frac{1}{2}x + 2$ Here:

• Slope ($m$) = $\frac{1}{2}$
• Y-intercept ($b$) = $2$

We can plot this line by starting at $(0, 2)$ (the y-intercept) and using the slope to move:

• Up 1 unit and right 2 units (slope = $\frac{1}{2}$).

Step 2: Graph the lines

1. Plot the line $y = -x - 4$:

   • Start at $(0, -4)$ (y-intercept).
   • Use the slope ($-1$): move down 1 unit and right 1 unit to get another point, like $(1, -5)$.
   • Draw the line through these points.

2. Plot the line $y = \frac{1}{2}x + 2$:

   • Start at $(0, 2)$ (y-intercept).
   • Use the slope ($\frac{1}{2}$): move up 1 unit and right 2 units to get another point, like $(2, 3)$.
   • Draw the line through these points.

Step 3: Find the Intersection Point

The intersection point of these two lines represents the solution to the system of equations.

By solving graphically, you should find the point where these two lines cross. Alternatively, solving algebraically:

1. Set $y = -x - 4$ equal to $y = \frac{1}{2}x + 2$: $-x - 4 = \frac{1}{2}x + 2$
2. Solve for $x$: $-\frac{3}{2}x = 6 \quad \Rightarrow \quad x = -4$
3. Solve for $y$ by substituting $x = -4$ into $y = -x - 4$: $y = -(-4) - 4 = 4 - 4 = 0$

Thus, the solution is: $x = -4, \, y = 0$

Would you like me to plot this graph for you or explain further?

Relative Questions

1. What are the steps to convert a linear equation into slope-intercept form?
2. How do you interpret the slope and y-intercept of a linear equation?
3. How can you verify solutions to systems of equations algebraically?
4. What are the differences between solving systems graphically and algebraically?
5. How do parallel lines behave in systems of equations?

Tip

When solving graphically, always double-check by substituting the solution into both original equations to confirm accuracy!