Question

1. sally and ray work together on an assignment. they have broken down of the same size. sally works at 2 times the speed of ray but has 5 piece another subject. this problem can be represented by the following system of equations: equation 1: $x + y = 4$ equation 2: $y = 2x - 5$ 6a sketch the two lines representing these equations on the graph below:

Explanation:

Step1: Analyze Equation 1 ($x + y = 4$)

Rewrite it in slope - intercept form ($y=mx + b$), we get $y=-x + 4$. The slope ($m$) is $- 1$ and the y - intercept ($b$) is $4$. To find two points on this line:

• When $x = 0$, $y=4$. So the point is $(0,4)$.
• When $y = 0$, $x = 4$. So the point is $(4,0)$.

Step2: Analyze Equation 2 ($y = 2x-5$)

The slope ($m$) is $2$ and the y - intercept ($b$) is $- 5$. To find two points on this line:

• When $x = 0$, $y=-5$. So the point is $(0, - 5)$.
• When $x = 2$, $y=2\times2 - 5=-1$. So the point is $(2,-1)$. When $x = 3$, $y=2\times3-5 = 1$. When $x = 4$, $y=2\times4 - 5=3$.

Step3: Sketch the lines

For $y=-x + 4$, plot the points $(0,4)$ and $(4,0)$ and draw a straight line through them. For $y = 2x-5$, plot the points (e.g., $(0,-5)$ and $(2,-1)$ or $(3,1)$) and draw a straight line through them. The two lines should intersect at the solution of the system of equations. To find the intersection, substitute $y = 2x-5$ into $x + y=4$:
$x+(2x - 5)=4$
$3x-5 = 4$
$3x=9$
$x = 3$
Then $y=2\times3-5 = 1$. So the intersection point is $(3,1)$. When sketching, the line $y=-x + 4$ has a negative slope and the line $y = 2x-5$ has a positive slope.

(Note: Since the problem asks to sketch the lines, the key is to identify the correct points for each line and draw them. The graph provided in the question has some blue dots, and we can verify that for $y=-x + 4$, when $x = 1$, $y = 3$; when $x = 2$, $y = 2$ etc. For $y=2x-5$, when $x = 3$, $y = 1$; when $x = 2$, $y=-1$ etc. The two lines should be drawn such that one has a slope of - 1 and the other has a slope of 2, intersecting at $(3,1)$.)

Answer:

(The process of sketching the lines: For $y=-x + 4$, use points like $(0,4)$ and $(4,0)$; for $y = 2x-5$, use points like $(0,-5)$ and $(3,1)$. Draw two straight lines through their respective points. The intersection point is $(3,1)$.) The sketch should show two lines, one with a negative slope (for $x + y = 4$) and one with a positive slope (for $y=2x - 5$) intersecting at $(3,1)$.