Question

graph the system of equations to find the solution. click \check my graph\ to see how you did!
$y = 3x + 1$
$y = -\frac{1}{2}x + 1$

Explanation:

Step1: Analyze the first equation

The first equation is \( y = 3x + 1 \). 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 \( m = 3 \) (a positive slope, so the line will rise from left to right) and the y - intercept \( b = 1 \), so the line passes through the point \( (0,1) \). To find another point, we can use the slope. For example, when \( x = 1 \), \( y=3(1)+1=4 \), so the line passes through \( (1,4) \) as well.

Step2: Analyze the second equation

The second equation is \( y=-\frac{1}{2}x + 1 \). This is also in slope - intercept form. The slope \( m=-\frac{1}{2} \) (a negative slope, so the line will fall from left to right) and the y - intercept \( b = 1 \), so this line also passes through the point \( (0,1) \). To find another point, using the slope: when \( x = 2 \), \( y=-\frac{1}{2}(2)+1=- 1 + 1=0 \), so the line passes through \( (2,0) \).

Step3: Find the intersection point

Since both lines have a y - intercept of 1 (they both pass through \( (0,1) \)), we can also solve the system of equations algebraically to confirm. Set the two equations equal to each other:
\( 3x + 1=-\frac{1}{2}x + 1 \)
Subtract 1 from both sides: \( 3x=-\frac{1}{2}x \)
Add \( \frac{1}{2}x \) to both sides: \( 3x+\frac{1}{2}x = 0 \)
\( \frac{6x + x}{2}=0\) (finding a common denominator)
\( \frac{7x}{2}=0 \)
Multiply both sides by \( \frac{2}{7} \): \( x = 0 \)
Substitute \( x = 0 \) into \( y = 3x+1 \), we get \( y=3(0)+1 = 1 \). So the solution to the system (the point of intersection of the two lines) is \( (0,1) \).

When graphing:

• For \( y = 3x + 1 \), plot the points \( (0,1) \) and \( (1,4) \) (or other points found using the slope) and draw a line through them.
• For \( y=-\frac{1}{2}x + 1 \), plot the points \( (0,1) \) and \( (2,0) \) (or other points found using the slope) and draw a line through them. The two lines will intersect at \( (0,1) \).

Answer:

The solution to the system of equations (the point of intersection of the two lines when graphed) is \( (0,1) \)