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Luke started a weight-loss program. The first week, he lost x pounds. The second week, he lost pounds less than times the pounds he lost the first week. The third week, he lost 1 pound more than of the pounds he lost the first week.

Liam started a weight-loss program when Luke did. The first week, he lost 1 pound less than times the pounds Luke lost the first week. The second week, he lost 4 pounds less than times the pounds Luke lost the first week. The third week, he lost pound more than times the pounds Luke lost the first week.

Assuming they both lost the same number of pounds over the three weeks, complete the following sentences.

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MrRoyal

The weight loss illustrates the use of algebraic equations

  • Luke's equation is [tex]\mathbf{Total = \frac{13x}{4} - \frac{1}{2} }[/tex].
  • Liam's equation is [tex]\mathbf{Total =\frac{21}{4}x - \frac{9}{2} }[/tex]
  • The value of x is [tex]\mathbf{x =2 }[/tex]
  • The total weight loss is [tex]\mathbf{Total =6 }[/tex]

Luke

The number of pounds he lost in each week is as follows

[tex]\mathbf{Week\ 1: x}[/tex]

[tex]\mathbf{Week\ 2: \frac 32x - \frac 32}[/tex] --- 3/2 less than 3/2 of first week

[tex]\mathbf{Week\ 3:1 + \frac 34x}[/tex] --- 1 more than 3/4 of first week

So, the total pound he lost is:

[tex]\mathbf{Total = x + \frac 32x - \frac 32 + 1 + \frac 34x }[/tex]

Collect like terms

[tex]\mathbf{Total = x + \frac 32x + \frac 34x- \frac 32 + 1 }[/tex]

Take LCM

[tex]\mathbf{Total = \frac{4x + 6x + 3x}{4} + \frac{2 - 3}{2} }[/tex]

[tex]\mathbf{Total = \frac{13x}{4} - \frac{1}{2} }[/tex]

Liam

The number of pounds he lost in each week is as follows

[tex]\mathbf{Week\ 1: \frac 32x - 1}[/tex] --- 1 less than 3/2 of what Luke lost in the first week

[tex]\mathbf{Week\ 2: \frac 52x - 4}[/tex] --- 4 less than 5/2 of what Luke lost in the first week

[tex]\mathbf{Week\ 3:\frac 54x + \frac 12}[/tex] --- 1 /2more than 5/4 of what Luke lost in the first week

So, the total pound he lost is:

[tex]\mathbf{Total =\frac 32x - 1 + \frac 52 x - 4 + \frac 54x +\frac12 }[/tex]

Collect like terms

[tex]\mathbf{Total =\frac 32x + \frac 52 x + \frac 54x- 1 - 4 +\frac12 }[/tex]

Take LCM

[tex]\mathbf{Total =\frac{6+10+5}{4}x - \frac{2 + 8 -1}{2} }[/tex]

[tex]\mathbf{Total =\frac{21}{4}x - \frac{9}{2} }[/tex]

The value of x

Since they lost the same number of pound, then:

[tex]\mathbf{Total = \frac{13x}{4} - \frac{1}{2} =\frac{21}{4}x - \frac{9}{2} }[/tex]

The value of x is calculated as follows:

[tex]\mathbf{\frac{13x}{4} - \frac{1}{2} = \frac{21}{4}x - \frac{9}{2} }[/tex]

Multiply through by 4

[tex]\mathbf{ 13x - 2 = 21x - 18 }[/tex]

Collect like terms

[tex]\mathbf{21x - 13x =18 - 2 }[/tex]

[tex]\mathbf{8x =16 }[/tex]

Divide both sides by 8

[tex]\mathbf{x =2 }[/tex]

Total weight loss by each

We have:

[tex]\mathbf{Total =\frac{21}{4}x - \frac{9}{2} }[/tex]

Substitute [tex]\mathbf{x =2 }[/tex]

[tex]\mathbf{Total =\frac{21}{4} \times 2 - \frac{9}{2} }[/tex]

[tex]\mathbf{Total =\frac{21}{2} - \frac{9}{2} }[/tex]

Take LCM

[tex]\mathbf{Total =\frac{21- 9}{2} }[/tex]

[tex]\mathbf{Total =\frac{12}{2} }[/tex]

[tex]\mathbf{Total =6 }[/tex]

Read more about algebraic equations at:

https://brainly.com/question/953809

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