A rectangle is 4 inches longer than it is wide. It's perimeter is 28 inches. Write a system of two equations in two variables that describes this information. Then solve the system graphically.
So let the variable x=standard for all four sides of this rectangle, and let the variable w=# of additional inches to length. Let w=4
1. Draw a rectangle
2. label sides of rectangle used for width with "X"
3.Label sides with length(top and bottom), with X+W
4. Put variables into an equation, equaled to the total perimeter(28 inches) {x+w+x+w+x+x=28}
5. Set the number "4"( number of additional inches to the length) in place of w. {x+4+x+4+x+x=28}
6. Combine like terms {x+x+x+x=4x, and 4+4=8.
7. Set up simplified version: 4x+8=28
8. Subtract 8 from both sides to get 'X' alone.{4x=20}
9. Divide both sides by 4 to get 'x' alone. 4x\4==20\4
10. Solve: x=5
To write a system of equations to describe the information given, let's assign variables to the dimensions of the rectangle.
Let's say the width of the rectangle is 'w' inches. As stated, the length of the rectangle is 4 inches longer than the width, so it would be represented as 'w + 4' inches.
The perimeter of a rectangle is given by the formula: Perimeter = 2(width + length). Since the perimeter is given as 28 inches, we can write the first equation as:
2(w + (w + 4)) = 28
Simplifying this equation, we get:
2(2w + 4) = 28
4w + 8 = 28
4w = 28 - 8
4w = 20
w = 20/4
w = 5
Now that we have the value for the width, we can substitute it back into the equation for the length:
Length = w + 4 = 5 + 4 = 9
Therefore, the width of the rectangle is 5 inches and the length is 9 inches.
To solve the system graphically, we can plot the two equations on a coordinate plane. The first equation is linear, which can be represented as a straight line, and the second equation is the equation of a rectangle, which can be represented as a vertical line.
The first equation is:
2(w + (w + 4)) = 28
Simplifying:
2(2w + 4) = 28
4w + 8 = 28
4w = 20
w = 5
This equation can be rewritten as:
2w + 4 = 14
2w = 10
w = 5
So, one point on this line is (5, 14).
The second equation represents the vertical line x = 5 (the width of the rectangle), which means it will have infinite y values for each x coordinate of 5.
Plotting these equations on a coordinate plane, the point (5, 14) will lie on the line 2x + y = 14, and there will be a vertical line at x = 5.
To solve graphically, draw these two lines on a coordinate plane and the point where they intersect will give us the solution to the system of equations.