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3(2-m)=12-m

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To solve the equation \( 3(2 - m) = 12 - m \), follow these steps:

  1. Distribute the 3 on the left side: \[ 3 \cdot 2 - 3 \cdot m = 6 - 3m \] So the equation becomes: \[ 6 - 3m = 12 - m \]

  2. Next, let's rearrange the equation to isolate \( m \). We can add \( 3m \) to both sides: \[ 6 = 12 - m + 3m \] This simplifies to: \[ 6 = 12 + 2m \]

  3. Now, subtract 12 from both sides: \[ 6 - 12 = 2m \] So we have: \[ -6 = 2m \]

  4. Finally, divide both sides by 2: \[ m = \frac{-6}{2} = -3 \]

Thus, the solution to the equation is: \[ m = -3 \]

The length of a rectangle is four times its width. If the perimeter of the rectangle is 50 yd , find its length and width

Let the width of the rectangle be \( w \) yards. According to the problem, the length \( l \) is four times the width, so we can express the length as:

\[ l = 4w \]

The formula for the perimeter \( P \) of a rectangle is:

\[ P = 2l + 2w \]

Given that the perimeter is 50 yards, we can set up the equation:

\[ 2l + 2w = 50 \]

Now, substituting \( l = 4w \) into the perimeter equation:

\[ 2(4w) + 2w = 50 \]

This simplifies to:

\[ 8w + 2w = 50 \]

Combining like terms, we get:

\[ 10w = 50 \]

Now, divide both sides by 10 to find \( w \):

\[ w = \frac{50}{10} = 5 \text{ yards} \]

Now, using \( w \) to find \( l \):

\[ l = 4w = 4 \times 5 = 20 \text{ yards} \]

Thus, the dimensions of the rectangle are:

  • Width: \( 5 \) yards
  • Length: \( 20 \) yards
Mrs. Hill is designing a rectangular vegetable garden for her backyard. The width of the garden is 9 1/2 feet shorter than three times its length. The perimeter of the garden is 45 feet.

Let the length of the garden be \( l \) feet. According to the problem, the width \( w \) is defined as:

\[ w = 3l - 9.5 \]

The formula for the perimeter \( P \) of a rectangle is given by:

\[ P = 2l + 2w \]

We know that the perimeter is 45 feet:

\[ 2l + 2w = 45 \]

Now we can substitute the expression for \( w \) into the perimeter equation:

\[ 2l + 2(3l - 9.5) = 45 \]

Now distribute the 2:

\[ 2l + 6l - 19 = 45 \]

Combine like terms:

\[ 8l - 19 = 45 \]

Now, add 19 to both sides:

\[ 8l = 64 \]

Now divide by 8:

\[ l = 8 \]

Now that we have the length, we can find the width using the expression we defined earlier:

\[ w = 3l - 9.5 \] \[ w = 3(8) - 9.5 = 24 - 9.5 = 14.5 \]

Thus, the dimensions of Mrs. Hill's vegetable garden are:

  • Length: \( 8 \) feet
  • Width: \( 14.5 \) feet
I need the Equation

Certainly! Let's derive the equations step-by-step based on the information provided:

  1. Define the Variables:

    • Let \( l \) be the length of the garden (in feet).
    • Let \( w \) be the width of the garden (in feet).

  2. Translate the Problem into Equations:

    • According to the given information, the width is 9.5 feet shorter than three times the length: \[ w = 3l - 9.5 \]
    • The perimeter of a rectangle is given by the formula: \[ P = 2l + 2w \]
    • We know the perimeter of the garden is 45 feet, so: \[ 2l + 2w = 45 \]

  3. Substituting the Expression for Width:

    • Substitute \( w = 3l - 9.5 \) into the perimeter equation: \[ 2l + 2(3l - 9.5) = 45 \]

  4. Simplifying the Equation:

    • Distribute the 2: \[ 2l + 6l - 19 = 45 \]
    • Combine like terms: \[ 8l - 19 = 45 \]

  5. Rearranging to Solve for Length:

    • Add 19 to both sides: \[ 8l = 64 \]
    • Divide by 8: \[ l = 8 \]

  6. Finding Width:

    • Use the first equation to find \( w \): \[ w = 3(8) - 9.5 = 24 - 9.5 = 14.5 \]

Summary of Equations:

  1. Width Equation: \[ w = 3l - 9.5 \]
  2. Perimeter Equation: \[ 2l + 2w = 45 \]

After solving, we find:

  • Length \( l = 8 \) feet
  • Width \( w = 14.5 \) feet
Just give me one Equation

Absolutely! Here’s a single equation that captures the relationship between the length \( l \) and the width \( w \) based on the conditions you provided (width being 9.5 feet shorter than three times the length and the perimeter being 45 feet):

\[ 2l + 2(3l - 9.5) = 45 \]

This equation can be solved to find the values of \( l \) and \( w \).