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 .
This expression involves the operations of multiplication and addition.
We will use the distributive property to simplify this expression.
The simplified expression looks like -6x - 21.
To solve the equation: 24(34+x)=1,200
, give the reasons for each of the given steps.
What was done to the previous step to get to the current equation?
(1 point)
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Step 1: 816+24x=1,200
Step 2: 816−816+24x=1,200 − 816
Step 3: 24x=384
Step 4: x=16
Step 1: 816+24x=1,200
Explanation: In the previous step, we multiplied 24 by the expression (34 + x) using the distributive property.
Step 2: 816−816+24x=1,200 − 816
Explanation: In this step, we simplified the left side of the equation by subtracting 816 from both sides. This was done to isolate the term with x on one side of the equation.
Step 3: 24x=384
Explanation: In this step, we further simplified the equation by combining like terms on the left side. We subtracted 816 from 816, which resulted in 0. The equation then becomes 24x = 384.
Step 4: x=16
Explanation: In the final step, we solved for x by dividing both sides of the equation by 24, which gives x = 16.
Put the steps in order for solving the equation: −2(4x+8)+2x = −5x+10
(1 point)
Step 1: Distribute the -2 to the terms inside the parentheses:
-8x - 16 + 2x = -5x + 10
Step 2: Combine like terms on each side of the equation:
-6x - 16 = -5x + 10
Step 3: Add 5x to both sides of the equation to isolate the terms with x on one side:
-x - 16 = 10
Step 4: To solve for x, add 16 to both sides of the equation:
-x = 26
Step 5: Multiply both sides of the equation by -1 to solve for x:
x = -26
The correct order of steps for solving the equation is:
1. Distribute the -2
2. Combine like terms
3. Add 5x to both sides
4. Add 16 to both sides
5. Multiply both sides by -1
A student solves the following problem: 2(x - 2) + 5x = 24
Step 1: 2x - 4 + 5x = 24
Step 2: 10x - 4 = 24
Step 3: 10x - 4 + 4 = 24 + 4
Step 4: 10x = 28
Step 5: 10x/10=28/10
Step 6: x = 2.8
Where is the mistake? What did the student do incorrectly?
(1 point)
Responses
A. Step 3: The student should have subtracted 4 from both sides, not added 4.
B. Step 1: The student should have only distributed the 2 and x, not the x & -2.
C. Step 2: The student should have added 2x + 5x = 7x, not (2x)(5) = 10x.
D. Step 5: The student should have multiplied by 10 on both sides, not divided by 10.
The mistake occurred in:
D. Step 5: The student should have multiplied by 10 on both sides, not divided by 10.
In Step 5, the student should have multiplied both sides of the equation by 10 to isolate the variable x. Dividing by 10 is incorrect and leads to an incorrect solution. The correct step should be:
10x = 280
Dividing both sides by 10 would result in:
x = 28
Therefore, the student made a mistake in Step 5 by dividing by 10 instead of multiplying by 10.