A string is cut in two so that one section is 5 inches longer than the other. Twice the length of the longer section is three times the length of the shorter section. Write an equation that allows you to solve for the length of the shorter section. How long is the shorter section?
a. 20 inches
b. 12 inches
c. 10 inches
d. 15 inches
Let's assume the length of the shorter section is x inches.
According to the given information, the length of the longer section is 5 inches longer than the shorter section. So, the length of the longer section is x + 5 inches.
Twice the length of the longer section is three times the length of the shorter section. In equation form, this can be written as:
2(x + 5) = 3x
Simplifying this equation, we get:
2x + 10 = 3x
Subtracting 2x from both sides, we get:
10 = x
Therefore, the length of the shorter section is 10 inches.
Therefore, the answer is c. 10 inches
Andrea is 6 years younger than Tony. In two years, Tony will be three times as old as Andrea. Let a be Tony’s current age. Which of these linear equations can be used to find Andrea’s and Tony’s current ages?(1 point)
Responses
A. a=3(a+2)
B. 3(a+2)=a−6+2
C. a+2=3(a−6+2)
D. a+2=3(a+6+2)
Let's analyze the given information step by step:
1. Andrea is 6 years younger than Tony. Therefore, we can express Andrea's age as a - 6.
2. In two years, Tony will be three times as old as Andrea. If a is Tony's current age, in two years his age will be a + 2. Andrea's age in two years will be (a - 6) + 2.
According to the given information, we can create the equation:
a + 2 = 3((a - 6) + 2)
Simplifying this equation, we get:
a + 2 = 3(a - 4)
Therefore, the correct equation to find Andrea's and Tony's current ages is:
C. a + 2 = 3(a - 4)
Five less than three-fourths of x is seven. If x equals 16, what is the last property of equality you would use to justify the solution to this problem?
A. Addition Property of Equality
B. Identity Property of Multiplication
C. Division Property of Equality
D. Subtraction Property of Equality
The last property of equality you would use to justify the solution to this problem is:
D. Subtraction Property of Equality
To solve the equation, "Five less than three-fourths of x is seven," you would start by translating the problem into an equation.
The equation would be:
(3/4)x - 5 = 7
To isolate the variable x, you would use the Subtraction Property of Equality. By adding 5 to both sides of the equation, you would get:
(3/4)x = 12
Then, you would need to solve for x by multiplying both sides of the equation by the reciprocal of 3/4 (which is 4/3):
(4/3)(3/4)x = (4/3)(12)
This simplifies to:
x = 16
Therefore, the last property of equality you would use to justify the solution to this problem is the Subtraction Property of Equality.
Manipulate the equation y=7x−12 so that it is written in terms of x. Which equation is equivalent to y=7x−12?(1 point)
Responses
A. x=(y7)+12
B. x=(y−12)7
C. x=(y+12)7
D. x=(y12)−7
To manipulate the equation y = 7x - 12 and write it in terms of x, we need to solve for x.
First, let's isolate the term containing x. We can do this by adding 12 to both sides of the equation:
y + 12 = 7x
Next, let's divide both sides of the equation by 7 to solve for x:
(y + 12) / 7 = x
Therefore, the equation equivalent to y = 7x - 12 in terms of x is:
C. x = (y + 12) / 7
Which equation is equivalent to w = x + y - z?
A. z = x + y - w
B. z = w - x - y
C. z = (x + y) / w
D. z = x + y + w
To obtain an equivalent equation to w = x + y - z, we can start by isolating z on one side of the equation.
First, let's subtract x and y from both sides:
w - x - y = -z
To make z the subject of the equation, we must multiply both sides of the equation by -1 to change the sign:
-(w - x - y) = z
Simplifying the equation further, we have:
-z = -w + x + y
And by multiplying both sides of the equation by -1 again to change the sign, we get:
z = w - x - y
Therefore, the equation equivalent to w = x + y - z is:
B. z = w - x - y
Look at the expression: −3(2x+7)
(1 point)
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This expression involves these two operations what and what .
We will use the what to simplify this expression.
The simplified expression looks like what .