Question

skill practice - slope-intercept form
a1: $y = 2x + 4$ and $y = -x + 1$
solve by graphing.

Explanation:

Step1: Analyze the first equation \( y = 2x + 4 \)

The slope - intercept form of a line is \( y=mx + b \), where \( m \) is the slope and \( b \) is the y - intercept. For the equation \( y = 2x+4 \), the slope \( m_1 = 2=\frac{2}{1}\) and the y - intercept \( b_1 = 4 \). To graph this line, we start by plotting the y - intercept at the point \( (0,4) \). Then, using the slope (rise over run), from the point \( (0,4) \), we move up 2 units (because the numerator of the slope is 2) and then 1 unit to the right (because the denominator of the slope is 1) to get the next point \( (1,6) \). We can also move down 2 units and 1 unit to the left from \( (0,4) \) to get the point \( (- 1,2) \). Then we draw a line through these points.

Step2: Analyze the second equation \( y=-x + 1 \)

For the equation \( y=-x + 1 \), the slope \( m_2=-1=\frac{-1}{1}\) and the y - intercept \( b_2 = 1 \). We start by plotting the y - intercept at the point \( (0,1) \). Using the slope, from the point \( (0,1) \), we move down 1 unit (because the numerator of the slope is - 1) and 1 unit to the right to get the point \( (1,0) \), or we can move up 1 unit and 1 unit to the left to get the point \( (-1,2) \). Then we draw a line through these points.

Step3: Find the intersection point

The solution to the system of equations (the point where the two lines intersect) is the point that satisfies both equations. By looking at the graphs of the two lines, we can see that the two lines intersect at the point \( (-1,2) \). We can also verify this algebraically by setting the two equations equal to each other:
\( 2x + 4=-x + 1 \)
Add \( x \) to both sides: \( 2x+x+4=-x+x + 1\)
\( 3x+4 = 1 \)
Subtract 4 from both sides: \( 3x+4 - 4=1 - 4\)
\( 3x=-3 \)
Divide both sides by 3: \( x = - 1 \)
Substitute \( x=-1 \) into one of the original equations, say \( y=-x + 1 \): \( y=-(-1)+1=1 + 1=2 \)

Answer:

The solution of the system of equations \( y = 2x + 4 \) and \( y=-x + 1 \) is \( x=-1,y = 2 \) or the ordered pair \( (-1,2) \)