Arthur writes the equation a=10b−7. He wants to rewrite it in terms of b. In Arthur’s first step, what value should he add to both sides of the equation?(1 point)
Arthur should add 7 to both sides of the equation.
Tilda writes the equation 5y=20x+8. She wants to rewrite the equation in terms of x. After subtracting 8 from both sides of the equation, what number should she divide both sides of the equation by in order to accomplish this goal?(1 point)
Tilda should divide both sides of the equation by 20 in order to rewrite it in terms of x.
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?(1 point)
Let's call the length of the shorter section x inches.
According to the problem, the longer section is 5 inches longer than the shorter section, so its length is x + 5 inches.
Twice the length of the longer section is three times the length of the shorter section, so we can write the equation as:
2(x + 5) = 3x
Now, let's solve this equation to find the value of x:
2x + 10 = 3x
10 = 3x - 2x
10 = x
So, the shorter section is 10 inches long.
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?
Let's solve this problem step by step:
Let a be Tony's current age. This means Andrea's current age can be represented as a - 6.
In two years, Tony's age will be a + 2, and Andrea's age will be (a - 6) + 2, which simplifies to a - 4.
According to the problem, Tony's age in two years will be three times Andrea's age in two years, so we can write the equation as:
a + 2 = 3(a - 4)
Now, let's simplify this equation:
a + 2 = 3a - 12
2 + 12 = 3a - a
14 = 2a
To find Andrea's current age, substitute the value of a/2 back into the equation a - 6.
(a - 6) - 6 = a - 12
Therefore, the linear equation that can be used to find Andrea's and Tony's current ages is:
a - 12 = 14 - 12
Simplifying this equation, we get:
a - 12 = 2
Therefore, the equation a - 12 = 2 can be used to find Andrea's and Tony's current ages.
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?(1 point)
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)
To rewrite the equation y = 7x - 12 in terms of x, we need to isolate x on one side of the equation.
Adding 12 to both sides of the equation, we have:
y + 12 = 7x
Now, we divide both sides of the equation by 7 to solve for x:
(y + 12)/7 = x
Therefore, the equivalent equation to y = 7x - 12 in terms of x is:
x = (y + 12)/7.
Which equation is equivalent to w=x+y−z?(1 point)
To find the equivalent equation to w = x + y - z, we need to manipulate it to isolate a variable on one side of the equation.
First, we can rearrange the equation to:
w + z = x + y
Next, we can subtract y from both sides of the equation:
w + z - y = x
Therefore, the equivalent equation to w = x + y - z is:
x = w + z - y.
To solve the equation: 24(34+x)=1,200, give the reasons for each of the given steps.
Step 1: Distribute 24
24(34 + x) = 1,200
Reason: We distribute 24 to both terms inside the parentheses to simplify the equation. This is done by multiplying each term inside the parentheses by 24.
Step 2: Simplify the equation
816 + 24x = 1,200
Reason: When we distribute 24, 24 multiplied by 34 is 816.
Step 3: Subtract 816 from both sides
816 + 24x - 816 = 1,200 - 816
Reason: We perform this step to isolate the term with x on one side of the equation.
Step 4: Simplify the equation
24x = 384
Reason: When we subtract 816 from 816, the result is 0, so it cancels out. When we subtract 816 from 1,200, the result is 384.
Step 5: Divide both sides by 24
(24x)/24 = 384/24
Reason: To solve for x, we divide both sides of the equation by 24 to isolate x and find its value.
Step 6: Simplify the equation
x = 16
Reason: When we divide (24x)/24, the 24's cancel out, leaving x = 16.
Therefore, the solution to the equation 24(34 + x) = 1,200 is x = 16.
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?
The mistake occurs in Step 3.
In Step 3, the student applies the operation of adding 4 to both sides of the equation. However, they made an error in the calculation.
The correct calculation would be: 10x - 4 + 4 = 24 + 4
Simplifying both sides of the equation correctly:
10x = 28
Therefore, the value of x is correctly determined in Step 6 as x = 2.8.
In summary, the student made an error in the calculation in Step 3, but the final result is still correct.
Put the steps in order for solving the equation: −2(4x+8)+2x = −5x+10(1 point)
To solve the equation:
-2(4x + 8) + 2x = -5x + 10
Here are the steps in the correct order:
Step 1: Distribute -2 to both terms in the parentheses.
-8x - 16 + 2x = -5x + 10
Step 2: Combine like terms on the left side of the equation.
-8x + 2x - 16 = -5x + 10
Step 3: Combine like terms on the right side of the equation.
-6x - 16 = -5x + 10
Step 4: Move all terms involving x to one side of the equation.
-6x + 5x - 16 = 10
Step 5: Combine like terms on the left side of the equation.
-x - 16 = 10
Step 6: Add 16 to both sides.
-x - 16 + 16 = 10 + 16
Step 7: Simplify the equation.
-x = 26
Step 8: Multiply both sides by -1 to isolate x.
(-1)(-x) = (-1)(26)
Step 9: Simplify the equation.
x = -26
Therefore, the correct order of steps to solve the equation is:
Step 1: Distribute -2 to both terms.
Step 2: Combine like terms on the left side of the equation.
Step 3: Combine like terms on the right side of the equation.
Step 4: Move all terms involving x to one side of the equation.
Step 5: Combine like terms on the left side of the equation.
Step 6: Add 16 to both sides.
Step 7: Simplify the equation.
Step 8: Multiply both sides by -1 to isolate x.
Step 9: Simplify the equation.
The solution to the equation is x = -26.