9514 1404 393
4) (x, y) = (1, 3)
5) length = 11 meters, width = 7 meters
Step-by-step explanation:
4) "Solve by graphing" means graph the two equations and find the coordinates of the point where they intersect.
For graphing these equations, it works well to start at the y-intercept and use the slope to find another point on the line.
These equations are in "slope-intercept form":
y = mx + b . . . . . . . . . where m is the slope and b is the y-intercept
In the first equation, the slope is 3 and the y-intercept is 0. (We rarely show +0 in an expression or equation.) This means the point where y=0 on the y-axis is one point on the line.
The slope is the ratio of "rise" to "run". A slope of 3 means the line will go up 3 grid squares for each 1 grid square to the right. That is, another point on the line is the point (1, 3).
To graph the first equation, draw a line through the two points (0, 0) and (1, 3).
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The second equation has a slope of 4 and a y-intercept of -1. This means the point y=-1 on the y-axis is a point on the line. The slope of 4 means it goes up 4 squares for 1 square to the right, so (x, y) = (1, -1+4) = (1, 3) will be another point on the graph of the second equation. To graph it, draw a line through the two points (0, -1) and (1, 3).
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If you're paying attention, you have figured out that the point (1, 3) is on both lines. That is the only point on both lines, so it is the solution to the system of equations. The solution is (x, y) = (1, 3). The graph is shown in the attachment.
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5) I find it convenient to solve this sort of problem using a single variable. Let w represent the width of the rectangle in meters. Then the length is (w+4), a length that is 4 meters longer than the width.
The perimeter is twice the sum of length and width. An equation expressing that is ...
P = 2(L +W)
36 = 2((w+4) +w) . . . . . . . . substitute known values into the formula
Simplifying the equation gives ...
36 = 2(2w +4)
36 = 4w +8
Subtracting 8 from both sides, we have ...
28 = 4w
Then we can divide both sides by 4 to find w.
7 = w . . . . . . the width
(w+4) = (7+4) = 11 . . . . . the length
The length and width of the rectangle are 11 meters and 7 meters, respectively.
