Question

solve the system of linear equations by graphing.
\

$$\begin{cases} y = -3x - 5 y = -x - 1 \\end{cases}$$

(graph with x and y axes, grid, and two lines with blue points)
enter the solution as an ordered pair in the form (x, y).

Explanation:

Step1: Recall the solution of a system of linear equations by graphing

The solution to a system of linear equations is the point of intersection of their graphs (the ordered pair \((x,y)\) that satisfies both equations).

Step2: Analyze the given equations

We have two linear equations:

• \(y = -3x - 5\)
• \(y = -x - 1\)

To find the intersection point, we can also solve the system algebraically (as a check) by setting the two expressions for \(y\) equal to each other:

$$ -3x - 5=-x - 1 $$

Step3: Solve for \(x\)

Add \(3x\) to both sides:

$$ -5 = 2x - 1 $$

Add \(1\) to both sides:

$$ -4 = 2x $$

Divide both sides by \(2\):

$$ x=-2 $$

Step4: Solve for \(y\)

Substitute \(x = -2\) into one of the equations, say \(y=-x - 1\):

$$ y=-(-2)-1=2 - 1 = 1 $$

Alternatively, from the graph, we can identify the point where the two lines cross. Looking at the graph, the intersection point appears to be \((-2, -1)\)? Wait, no, wait. Wait, when \(x = -2\), let's check \(y=-3x - 5\): \(y=-3(-2)-5 = 6 - 5 = 1\). And \(y=-x - 1\): \(y=-(-2)-1 = 2 - 1 = 1\). Wait, maybe my initial graph reading was wrong. Wait, let's re - check the equations.

Wait, the first equation is \(y=-3x - 5\). Let's find two points on this line:

• When \(x = 0\), \(y=-5\) (so the y - intercept is \((0,-5)\))
• When \(y = 0\), \(0=-3x - 5\Rightarrow3x=-5\Rightarrow x=-\frac{5}{3}\approx - 1.67\)

The second equation is \(y=-x - 1\):

• When \(x = 0\), \(y=-1\) (y - intercept \((0,-1)\))
• When \(y = 0\), \(0=-x - 1\Rightarrow x=-1\)

Now, when we solve \( - 3x-5=-x - 1\):

$$ -3x+x=-1 + 5 $$

$$ -2x = 4 $$

$$ x=-2 $$

Then \(y=-(-2)-1=2 - 1 = 1\). Wait, so the intersection point is \((-2,1)\)? Wait, maybe the graph in the problem has a typo or my mis - reading. Wait, let's re - examine the equations again.

Wait, the first equation: \(y=-3x - 5\). Let's plug \(x=-2\): \(y=-3\times(-2)-5=6 - 5 = 1\). Second equation: \(y=-x - 1\), \(x = - 2\), \(y=-(-2)-1=2 - 1 = 1\). So the solution is \((-2,1)\). But let's check the graph again. If the two lines intersect at \((-2, - 1)\), that would be wrong. Wait, maybe I made a mistake in the algebraic solution. Wait, no:

Wait, \(y=-3x - 5\) and \(y=-x - 1\). Set equal:

\(-3x-5=-x - 1\)

\(-3x+x=-1 + 5\)

\(-2x = 4\)

\(x=-2\)

Then \(y=-(-2)-1=1\). Yes, so the solution is \((-2,1)\).

Answer:

\((-2,1)\)