Following the steps provided:

1. Read and understand: we are looking for a number.
2. Constants and variables: 28 and 232 are constants, “a certain number” is our variable because we don’t know its value, and we are asked to find it. We will call it x.
3. Translate: five times a certain number translates to [latex]5x[/latex]
   Twenty-eight less than five times a certain number translates to [latex]5x-28[/latex] because subtraction is built backward.
   is 232 translates to [latex]=232[/latex] because “is” is associated with equals.
4. Write an equation: [latex]5x-28=232[/latex]
5. Solve the equation using what you know about solving linear equations:

   [latex]\begin{array}{r}5x-28=232\\5x=260\\x=52\,\,\,\end{array}[/latex]

6. Check and interpret: We can substitute 52 for x.

   [latex]\begin{array}{r}5\left(52\right)-28=232\\5\left(52\right)=260\\260=260\end{array}[/latex].

   TRUE!

Translate:

When there is a word that indicates multiplication just prior to a word that means either addition or subtraction, parentheses are needed.

Therefore, “three times the difference of a number and 4” translates into [latex]3(x-4)[/latex].  This is because three is being multiplied by the entire subtraction expression and not just [latex]x[/latex].

Write an equation:

Translating the entire sentence results in the following equation:  [latex]3(x-4)=18[/latex]

Solve the equation:

[latex]3(x-4)=18[/latex]

[latex]3x-12=18[/latex]

[latex]\underline{\hspace{.32in}+12 \hspace{.1in}+12}[/latex]

[latex]\hspace{.05in}3x\hspace{.42in}=30[/latex]

[latex]\frac{3x}{3}=\frac{30}{3}[/latex]

[latex]x=10[/latex]

Check:

[latex]3(x-4)=18[/latex]

[latex]3(10-4)=18[/latex]

[latex]3(6)=18[/latex]

[latex]18=18[/latex]

TRUE

Answer

The number is 10.

1. Read and understand: We are looking for three numbers, and we know they are consecutive integers.
2. Constants and Variables: 93 is a constant.
   The first integer we will call x.
   Second: [latex]x+1[/latex]
   Third: [latex]x+2[/latex]
3. Translate: The sum of three consecutive integers translates to [latex]x+\left(x+1\right)+\left(x+2\right)[/latex], based on how we defined the first, second, and third integers. Notice how we placed parentheses around the second and third integers. This is just to make each integer more distinct. is 93 translates to [latex]=93[/latex] because is is associated with equals.
4. Write an equation: [latex]x+\left(x+1\right)+\left(x+2\right)=93[/latex]
5. Solve the equation using what you know about solving linear equations: We can’t simplify within each set of parentheses, and we don’t need to use the distributive property so we can rewrite the equation without parentheses.

   [latex]x+x+1+x+2=93[/latex]

   Combine like terms, simplify, and solve.

   [latex]\begin{array}{r}x+x+1+x+2=93\\3x+3 = 93\\\underline{-3\,\,\,\,\,-3}\\3x=90\\\frac{3x}{3}=\frac{90}{3}\\x=30\end{array}[/latex]

6. Check and Interpret: Okay, we have found a value for x. We were asked to find the value of three consecutive integers, so we need to do a couple more steps. Remember how we defined our variables: The first integer we will call [latex]x[/latex], [latex]x=30[/latex]
   Second: [latex]x+1[/latex] so [latex]30+1=31[/latex]
   Third: [latex]x+2[/latex] so [latex]30+2=32[/latex] The three consecutive integers whose sum is [latex]93[/latex] are [latex]30\text{, }31\text{, and }32[/latex]

Let [latex]x=[/latex] the first even integer.      Then [latex]x+2=[/latex] the next consecutive even integer.

[latex]\begin{array}{r}x+(x+2)=-74\\2x+2 = -74\\\underline{-2\,\,\,\,\,-2}\\2x=-76\\\frac{2x}{2}=\frac{-76}{2}\\x=-38\\x+2=-36\end{array}[/latex]

The integers are -38 and -36.

Let [latex]x=[/latex] the first odd integer. Then [latex]x+2[/latex] and  [latex]x+4[/latex] would represent the next two consecutive odd integers.

[latex]\begin{array}{r}x+(x+2)+(x+4)=-15\\3x+6 = -15\\\underline{-6\,\,\,\,\,-6}\\3x=-21\\\frac{3x}{3}=\frac{-21}{3}\\x=-7\\x+2=-5\\x+4=-3\end{array}[/latex]

The integers are -7, -5 and -3

Let [latex]x=[/latex] the width of the rectangle. Then [latex]x+7[/latex] would represent how long the rectangle is.

Remember that opposite sides of a rectangle are equal in length.

[latex]\begin{array}{r}x+(x+7)+x + (x+7)=62\\4x+14 = 62\\\underline{-14\,\,-14}\\4x=48\\\frac{4x}{4}=\frac{48}{4}\\x=12\\x+7=19\end{array}[/latex]

The width of the room is 12 meters.

The length the room is 19 meters.

Let [latex]x=[/latex] the length of the shortest side.

[latex]\begin{array}{r}x+2x+4x =63\\7x= 63\\\frac{7x}{7}=\frac{63}{7}\\x=9\\2x=18\\4x=36\end{array}[/latex]

The lengths of the sides of the triangle are 9 ft., 18 ft., and 36 ft.

Let [latex]L=[/latex] the cost of the loveseat. Let [latex]2L=[/latex] the cost of the sofa. (Notice we use [latex]2L[/latex] here to indicate “twice the cost” and NOT [latex]L+2[/latex].)

Cost of the loveseat + Cost of the sofa  = 630

[latex]\begin{array}{r}L + 2L =630\\3L= 630\\\frac{3L}{3}=\frac{630}{3}\\L=210\\2L=420\end{array}[/latex]

The loveseat costs $210, and the sofa costs $420.

Let [latex]B=[/latex] the number of blue marbles. Let [latex]B+17=[/latex] the number of green marbles

Number of blue marbles + Number of green marbles  = 111

[latex]\begin{array}{r}B + (B+17) =111\\2B+17= 111\\\underline{-17\,\,\,-17}\\2B= 94\\\frac{2B}{2}=\frac{94}{2}\\B=47\\B+17=64\end{array}[/latex]

There are 47 blue marbles and 64 green marbles.

Notice how the final answers adds up to the total, 47 + 64 = 111 marbles.

Let [latex]x=[/latex] the length of the longer piece. Then [latex]x-4=[/latex] the length of the shorter piece.

Length of longer piece + Length of shorter piece = 12

[latex]\begin{array}{r}x + (x-4) =12\\2x-4= 12\\\underline{+4\,\,\,+4}\\2x= 16\\\frac{2x}{2}=\frac{16}{2}\\x=8\\x-4=4\end{array}[/latex]

The longer piece is 8 feet and the shorter piece is 4 feet.

Again, notice how the final answers add up to the total, 8 feet + 4 feet = 12 feet.