solve by graphing. \\begin{cases} y = 2x + 3 \\\\ y = x + 1 \\end{cases} (\\boxed{?}, \\boxed{})

Explanation:

Step1: Analyze the first line equation

The first equation is ( y = 2x + 3 ). This is a linear equation in slope - intercept form (( y=mx + b ), where ( m ) is the slope and ( b ) is the y - intercept). The slope ( m_1=2 ) and the y - intercept ( b_1 = 3 ). To graph this line, we start at the point ( (0,3) ) on the y - axis and then use the slope. Since the slope is ( 2=\frac{2}{1} ), we move up 2 units and to the right 1 unit from the y - intercept to get another point on the line, for example, ( (1,5) ).

Step2: Analyze the second line equation

The second equation is ( y=x + 1 ). It is also in slope - intercept form with slope ( m_2 = 1 ) and y - intercept ( b_2=1 ). We start at the point ( (0,1) ) on the y - axis and use the slope ( 1=\frac{1}{1} ) to find another point, such as ( (1,2) ).

Step3: Find the intersection point

To solve the system of equations by graphing, we need to find the point where the two lines intersect. Another way is to set the two equations equal to each other since at the intersection point, the ( y ) - values (and ( x ) - values) of the two equations are equal. Set ( 2x+3=x + 1 ). Subtract ( x ) from both sides: ( 2x - x+3=x - x + 1 ), which simplifies to ( x+3 = 1 ). Then subtract 3 from both sides: ( x+3-3=1 - 3 ), so ( x=- 2 ).

Step4: Find the value of y

Now that we have ( x=-2 ), we can substitute ( x = - 2 ) into either of the original equations. Let's use ( y=x + 1 ). Substitute ( x=-2 ) into ( y=x + 1 ), we get ( y=-2 + 1=-1 ).

Answer:

((-2,-1))