Question

solve the system by graphing. \\(\

$$\begin{cases} y = 2x + 3 \\\\ y = -4x - 3 \\end{cases}$$

\\)
solve the system by graphing. \\(\

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

\\)
your turn - (hint: use step 3 and check!)
without graphing, determine the solution of the system of linear equations. \\(\

$$\begin{cases} y = 2x - 1 \\\\ y = -x + 5 \\end{cases}$$

\\)
a \\((-2, -3)\\)
b \\((2, 3)\\)
c \\((-3, 2)\\)
d \\((3, 2)\\)
exit ticket
quick write: how can you use what you know about graphing a line to solve a system of linear equations?

Explanation:

First System (Graphing):

$$\begin{cases} y = 2x + 3 \\ y = -4x - 3 \end{cases}$$

Step 1: Analyze the first equation $y = 2x + 3$

This is a linear equation in slope - intercept form ($y=mx + b$), where the slope $m = 2$ and the y - intercept $b = 3$. To graph this line, we start at the y - intercept $(0,3)$ and then use the slope. Since the slope is $2=\frac{2}{1}$, we can go up 2 units and right 1 unit from the y - intercept to get another point, for example, $(1,5)$.

Step 2: Analyze the second equation $y=-4x - 3$

This is also in slope - intercept form with slope $m=-4$ and y - intercept $b = - 3$. We start at the y - intercept $(0,-3)$ and use the slope. Since the slope is $-4=\frac{-4}{1}$, we go down 4 units and right 1 unit from the y - intercept to get another point, for example, $(1,-7)$.

Step 3: Find the intersection point

When we graph both lines, we look for the point where they cross each other. To find the exact intersection, we can also solve the system algebraically (even though the problem says to solve by graphing, we can verify). Set $2x + 3=-4x - 3$.
Add $4x$ to both sides: $2x+4x + 3=-4x + 4x-3\Rightarrow6x + 3=-3$
Subtract 3 from both sides: $6x+3 - 3=-3 - 3\Rightarrow6x=-6$
Divide both sides by 6: $x=-1$
Substitute $x = - 1$ into $y = 2x+3$: $y=2\times(-1)+3=-2 + 3 = 1$. So the solution is $(-1,1)$.

Second System (Graphing):

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

Step 1: Analyze the first equation $y=\frac{1}{2}x$

This is a linear equation in slope - intercept form with slope $m=\frac{1}{2}$ and y - intercept $b = 0$ (it passes through the origin). To graph this line, we start at $(0,0)$ and use the slope $\frac{1}{2}$ (go up 1 unit and right 2 units) to get points like $(2,1)$.

Step 2: Analyze the second equation $y=x - 2$

This is in slope - intercept form with slope $m = 1$ and y - intercept $b=-2$. We start at the y - intercept $(0,-2)$ and use the slope (go up 1 unit and right 1 unit) to get points like $(1,-1)$.

Step 3: Find the intersection point

To find the intersection algebraically (to verify the graph), set $\frac{1}{2}x=x - 2$.
Subtract $\frac{1}{2}x$ from both sides: $0=x-\frac{1}{2}x-2\Rightarrow0=\frac{1}{2}x - 2$
Add 2 to both sides: $2=\frac{1}{2}x$
Multiply both sides by 2: $x = 4$
Substitute $x = 4$ into $y=\frac{1}{2}x$: $y=\frac{1}{2}\times4 = 2$. So the solution is $(4,2)$.

Your Turn (Without Graphing):

$$\begin{cases} y = 2x-1 \\ y=-x + 5 \end{cases}$$

Step 1: Set the two equations equal to each other

Since both equations are solved for $y$, we can set $2x-1=-x + 5$.

Step 2: Solve for $x$

Add $x$ to both sides: $2x+x-1=-x+x + 5\Rightarrow3x-1 = 5$
Add 1 to both sides: $3x-1 + 1=5 + 1\Rightarrow3x=6$
Divide both sides by 3: $x = 2$

Step 3: Solve for $y$

Substitute $x = 2$ into $y = 2x-1$ (we could also use the other equation). $y=2\times2-1=4 - 1=3$
So the solution is $(2,3)$, which corresponds to option B.

Exit Ticket (Quick Write)

Answer:

To solve a system of linear equations by graphing, graph each linear equation (using slope - intercept form for easy graphing). The solution is the point of intersection of the lines. Parallel lines have no solution, coincident lines have infinitely many solutions, and intersecting lines have one solution (the intersection point).

First System Solution (Graphing):

The solution of the system

$$\begin{cases}y = 2x + 3\\y=-4x - 3\end{cases}$$

is $\boldsymbol{(-1,1)}$

Second System Solution (Graphing):

The solution of the system

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

is $\boldsymbol{(4,2)}$

Your Turn Solution:

The solution of the system

$$\begin{cases}y = 2x-1\\y=-x + 5\end{cases}$$

is $\boldsymbol{B. (2, 3)}$

Exit Ticket Answer:

To solve a system of linear equations by graphing, graph each linear equation (using slope - intercept form for easy graphing). The solution is the point of intersection of the lines. Parallel lines have no solution, coincident lines have infinitely many solutions, and intersecting lines have one solution (the intersection point).