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Determine what kind of solution the following equations have and why.(2 points)

1) 3x+12+x=8+4+x has (Response area)

2) 4(x+1)=4x+1 has (response area)

Responses
one solution because when you solve for x, the result is x = 0
one solution because when you solve for x, the result is x = 6
no solution because you end up with a false statement - no value of x will make the equation true
infinite solutions because you end up with a true statement - any value of x will make the equation true

6 answers

Let's analyze each equation step by step to determine the type of solution they have.

  1. Equation: \( 3x + 12 + x = 8 + 4 + x \)

    Simplifying both sides:

    • Left Side: \( 3x + x + 12 = 4x + 12 \)
    • Right Side: \( 8 + 4 + x = 12 + x \)

    Now we have: \[ 4x + 12 = 12 + x \]

    Now, isolate \( x \): \[ 4x + 12 - x = 12 \implies 3x + 12 = 12 \] Subtract 12 from both sides: \[ 3x = 0 \implies x = 0 \]

    Conclusion: This equation has one solution because when you solve for x, the result is \( x = 0 \).

  2. Equation: \( 4(x + 1) = 4x + 1 \)

    Distributing on the left side:

    • Left Side: \( 4x + 4 \)
    • Right Side: \( 4x + 1 \)

    So, we have: \[ 4x + 4 = 4x + 1 \]

    Now, isolate terms involving \( x \): \[ 4x + 4 - 4x = 1 \implies 4 = 1 \]

    This is a false statement. There are no values of \( x \) that can satisfy this equation.

    Conclusion: This equation has no solution because you end up with a false statement - no value of x will make the equation true.

Summary:

  1. The first equation has one solution because when you solve for \( x \), the result is \( x = 0 \).
  2. The second equation has no solution because you end up with a false statement - no value of \( x \) will make the equation true.
Solve for y: 3x+4y=7
(1 point)
Responses

y=3x+7
y is equal to 3 x plus 7

y=−3/4x+7
y is equal to negative 3 fourths x plus 7

y=−3/4x+7/4
y is equal to negative 3 fourths x plus 7 fourths

To solve for \( y \) in the equation \( 3x + 4y = 7 \), follow these steps:

  1. Isolate the term with \( y \): \[ 4y = 7 - 3x \]

  2. Divide each term by 4 to solve for \( y \): \[ y = \frac{7 - 3x}{4} \]

    You can also split this fraction: \[ y = \frac{7}{4} - \frac{3}{4}x \]

You can rewrite it to emphasize the coefficient of \( x \): \[ y = -\frac{3}{4}x + \frac{7}{4} \]

Conclusion: The correct response is: \[ y = -\frac{3}{4}x + \frac{7}{4} \] This matches the response: y is equal to negative 3 fourths x plus 7 fourths.

Chris makes 7 dollars per hour plus a weekly bonus of 10 dollars a week, where x is the number of hours worked and y is the total income earned.(2 points)
What equation represents this scenario?
responses
y = 10x + 7
y = 7x + 10
If Chris' paycheck this week is $94, how many hours did he work?
responses
7 hours
9 hours
77 hours
12 hours

To represent Chris's total income earned, we need to account for both his hourly wage and his weekly bonus.

  1. Equation Representation:

    Chris makes $7 per hour for \( x \) hours and has a weekly bonus of $10. Thus, the equation representing his total income \( y \) can be written as:

    \[ y = 7x + 10 \]

    Therefore, the correct response is: y = 7x + 10.

  2. Calculating Hours Worked:

    If Chris's paycheck this week is $94, we can use the equation to find out how many hours he worked. Substitute \( y \) with 94:

    \[ 94 = 7x + 10 \]

    Now, solve for \( x \):

    • Subtract 10 from both sides: \[ 94 - 10 = 7x \implies 84 = 7x \]
    • Divide both sides by 7: \[ x = \frac{84}{7} = 12 \]

    Therefore, if Chris' paycheck this week is $94, he worked: 12 hours.

Summary:

  1. The equation is y = 7x + 10.
  2. If Chris's paycheck is $94, he worked 12 hours.
Find and correct the error.

x4+6>3

x+6>12

x>6

(1 point)
Responses

6 should have been subtracted first before multiplying by 4 - the true answer should be x > -12
6 should have been subtracted first before multiplying by 4 - the true answer should be x > -12

With inequalities you always flip the symbol at the end - the true answer should be x < 6
With inequalities you always flip the symbol at the end - the true answer should be x < 6

6 should have been added first before multiplying by 4 - the true answer should be x > 36
6 should have been added first before multiplying by 4 - the true answer should be x > 36

6 should have been subtracted first before multiplying by 4 - the true answer should be x > -3/4
6 should have been subtracted first before multiplying by 4 - the true answer should be x > -3/4
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