Solve and justify your answer using the properties of operations (i.e Distributive property, associative property, commutative property, identity property, substitution, ect…)
2x - 5 = 8x + 7
To solve the equation 2x - 5 = 8x + 7, we can use the properties of operations to isolate the variable x on one side of the equation.
First, let's simplify the equation by combining like terms. We can do this by adding 5 to both sides:
2x - 5 + 5 = 8x + 7 + 5
2x = 8x + 12
Next, let's get rid of the 8x term on the right side of the equation. We can do this by subtracting 8x from both sides:
2x - 8x = 8x + 12 - 8x
-6x = 12
Now, let's solve for x by dividing both sides of the equation by -6:
(-6x)/(-6) = 12/(-6)
x = -2
Therefore, the solution to the equation 2x - 5 = 8x + 7 is x = -2.
so which property was used? associative, commutative ?
In this particular problem, we did not use the associative or commutative properties.
The properties that were used are:
1. Addition/Subtraction Property: We used the addition property to add 5 to both sides of the equation and the subtraction property to subtract 8x from both sides of the equation.
2. Equality Property: We used the property of equality to state that if two expressions are equal, then adding or subtracting the same value from both sides of an equation will still result in equality.
3. Division Property: We used the division property to divide both sides of the equation by -6 in order to solve for x.
-9 + 6x = -3(3 - 2x)
To solve the equation -9 + 6x = -3(3 - 2x), we can use the properties of operations to simplify the equation and isolate the variable x.
First, let's simplify the right side of the equation by using the distributive property:
-3(3 - 2x) = -3 * 3 + (-3) * (-2x)
-3(3 - 2x) = -9 + 6x
Now, we can rewrite the equation with the simplified right side:
-9 + 6x = -9 + 6x
Next, let's combine like terms by subtracting 6x from both sides of the equation:
-9 + 6x - 6x = -9 + 6x - 6x
-9 = -9
The equation -9 = -9 is true for all values of x. This means that the equation has infinitely many solutions.
To justify this, we can use the reflexive property of equality, which states that any quantity is equal to itself. In this case, -9 is equal to -9, and therefore, the equation is true.
Therefore, the solution to the equation -9 + 6x = -3(3 - 2x) is x can be any real number.
P = 2L + 2W for W
To solve the equation P = 2L + 2W for W, we need to isolate W on one side of the equation. Let's follow the steps:
First, let's subtract 2L from both sides of the equation:
P - 2L = 2L + 2W - 2L
Simplifying the right side gives us:
P - 2L = 2W
Next, let's divide both sides of the equation by 2:
(P - 2L) / 2 = (2W) / 2
Simplifying further:
(P - 2L) / 2 = W
Therefore, we have the solution W = (P - 2L) / 2.
To justify this solution, we used the properties of operations:
1. Subtraction Property: We subtracted 2L from both sides in order to get the W term isolated on one side.
2. Division Property: We divided both sides of the equation by 2 to solve for W.
Therefore, our solution W = (P - 2L) / 2 is justified using the properties of operations.
ax + b = 0 for x
To solve the equation ax + b = 0 for x, we need to isolate x on one side of the equation. Let's follow the steps:
First, let's subtract b from both sides of the equation:
ax + b - b = 0 - b
Simplifying the equation gives us:
ax = -b
Next, let's divide both sides of the equation by a:
(ax) / a = (-b) / a
Simplifying further:
x = -b/a
Therefore, we have the solution x = -b/a.
To justify this solution, we used the properties of operations:
1. Subtraction Property: We subtracted b from both sides in order to isolate the ax term on one side.
2. Division Property: We divided both sides of the equation by a to solve for x.
Therefore, our solution x = -b/a is justified using the properties of operations.
V=LWH for H
To solve the equation V = LWH for H, we need to isolate H on one side of the equation. Let's follow the steps:
First, let's divide both sides of the equation by LW:
V / (LW) = (LWH) / (LW)
Simplifying the equation gives us:
V / (LW) = H
Therefore, we have the solution H = V / (LW).
To justify this solution, we used the properties of operations:
1. Division Property: We divided both sides of the equation by LW to isolate H on one side.
Therefore, our solution H = V / (LW) is justified using the properties of operations.
y=mx + b for x
To solve the equation y = mx + b for x, we can isolate x on one side of the equation. Let's follow the steps:
First, let's subtract b from both sides of the equation:
y - b = mx + b - b
Simplifying the equation gives us:
y - b = mx
Next, let's divide both sides of the equation by m:
(y - b) / m = (mx) / m
Simplifying further:
(y - b) / m = x
Therefore, we have the solution x = (y - b) / m.
To justify this solution, we used the properties of operations:
1. Subtraction Property: We subtracted b from both sides of the equation to isolate the mx term on one side.
2. Division Property: We divided both sides of the equation by m to solve for x.
Therefore, our solution x = (y - b) / m is justified using the properties of operations.
If the perimeter of a rectangle is 60ft and its length is 18ft, find its width.
To find the width of the rectangle, we can use the given information that the perimeter is 60ft and the length is 18ft.
We know that the perimeter of a rectangle is given by the formula:
Perimeter = 2 * (length + width)
Substituting the given values into the formula, we get:
60 = 2 * (18 + width)
Next, let's simplify the equation:
60 = 36 + 2 * width
Now, let's isolate the width term:
60 - 36 = 2 * width
24 = 2 * width
Finally, let's solve for width by dividing both sides of the equation by 2:
24/2 = (2 * width) / 2
12 = width
Therefore, the width of the rectangle is 12ft.
Next, let's simplify the equation:
60 = 36 + 2 * width
so where does this 2 come from?
Apologies for the confusion. You're right, the equation
60 = 36 + 2 * width
has a 2 in it, which comes from the fact that the formula for the perimeter of a rectangle is P = 2 * (length + width).
By distributing the 2, we obtain 2 * width, as both the length and width contribute to two sides of the rectangle.