Question

4 a. graph the following system of equations.
$y = -3x$
$y = x + 4$
b. what does the graph show to be the solution of the system?
5 deandre and his sister asha make origami cranes. their goal is to
complete 1,000 cranes by the end of the summer.

• deandre already has 30 cranes and makes 5 more each day.
• asha already has 10 cranes and makes 15 more each day.

the graph shows how many cranes, c, each person has made after d days.
a. what does the graph show to be the solution of the system?
b. what does the solution mean in this context?

Explanation:

Problem 4a

Step1: Analyze \( y = -3x \)

This is a linear equation in slope - intercept form \( y=mx + b \), where \( m=-3 \) (slope) and \( b = 0 \) (y - intercept). To graph it, we can find two points. When \( x = 0 \), \( y=0 \). When \( x = 1 \), \( y=-3(1)=-3 \). So we have points \((0,0)\) and \((1, - 3)\).

Step2: Analyze \( y=x + 4 \)

This is also in slope - intercept form with \( m = 1 \) (slope) and \( b=4 \) (y - intercept). When \( x = 0 \), \( y = 4 \). When \( x=-4 \), \( y=-4 + 4=0 \). So we have points \((0,4)\) and \((-4,0)\).

Step3: Graph the lines

Plot the points for each line and draw a straight line through them. The line for \( y=-3x \) will pass through the origin with a steep negative slope, and the line for \( y=x + 4 \) will cross the y - axis at \( (0,4) \) and the x - axis at \( (-4,0) \) with a positive slope.

Problem 4b

The solution of a system of linear equations is the point of intersection of their graphs. To find the intersection of \( y=-3x \) and \( y=x + 4 \), we can set the two equations equal to each other:

Step1: Set equations equal

\(-3x=x + 4\)

Step2: Solve for \( x \)

Subtract \( x \) from both sides: \(-3x-x=x + 4-x\)
\(-4x=4\)
Divide both sides by \(-4\): \(x=\frac{4}{-4}=-1\)

Step3: Find \( y \)

Substitute \( x=-1 \) into \( y=-3x \), we get \( y=-3(-1)=3 \)

So the solution is the point \((-1,3)\)

Problem 5a

The system of equations for the number of cranes:
For DeAndre: \( c = 5d+30 \) (since he starts with 30 cranes and makes 5 per day)
For Asha: \( c=15d + 10 \) (since she starts with 10 cranes and makes 15 per day)

The solution of the system is the point of intersection of the two lines in the graph. From the graph, we can see that the lines intersect at \( d = 2 \) and \( c = 40 \) (by looking at the x - coordinate (days) and y - coordinate (cranes) of the intersection point).

Problem 5b

Answer:

s:
4a. Graph with \( y=-3x \) (passing through \((0,0)\) and \((1, - 3)\)) and \( y=x + 4 \) (passing through \((0,4)\) and \((-4,0)\)) drawn.

4b. The solution is \( \boldsymbol{(-1,3)} \)

5a. The solution is \( \boldsymbol{(d = 2,c = 40)} \) (or \( (2,40) \) where \( d \) is days and \( c \) is cranes)

5b. After 2 days, both DeAndre and Asha will have made 40 origami cranes.