Linear Equations in Disguise

Linear Equations in Disguise exercise

• Identify the steps for solving the linear equation in disguise.

  Hints

  Below is an example of solving a linear equation in disguise, with the steps provided.

  $ \begin{array}{c|l} \text{solution} & \text{step}\\ \hline \\ \frac{x}{6}=\frac{3}{5} & \text{original equation}\\ \\ 5x=18 & \text{cross-multiplication}\\ \\ x=\frac{18}{5} & \text{division}\\ \\ x=2\frac35 & \text{conversion to mixed number} \end{array} $

  Below is an example of solving a linear equation in disguise, with the steps provided.

  $ \begin{array}{c|l} \text{solution} & \text{step}\\ \hline \\ \frac{x-7}{6}=\frac{3x+1}{5} & \text{original equation}\\ \\ 5(x-7)=6(3x+1) & \text{cross-multiplication}\\ \\ 5x-35=18x+6 & \text{distribution}\\ \\ -13x=41 & \text{combine like terms}\\ \\ x=-\frac{41}{13} & \text{division}\\ \\ x=-3\frac{2}{13} & \text{conversion to mixed number} \end{array} $

  Solution

  $ \begin{array}{c|l} \text{solution} & \text{step}\\ \hline \\ \frac{x+2}{3}=\frac{x-5}{9} & \text{original equation}\\ \\ 9(x+2)=3(x-5) & \text{cross-multiply}\\ \\ 9x+18=3x-15 & \text{distribution}\\ \\ 6x=-33 & \text{combine like terms}\\ \\ x=-\frac{33}{6} & \text{division}\\ \\ x=-\frac{11}{2} & \text{reduce the fraction}\\ \\ x=-5\frac12 & \text{convert to mixed number}\\ \end{array} $

• Solve the linear equations in disguise.

  Hints

  Here's an example of solving a linear equation in disguise.$ \begin{array}{c|l} \text{solution} & \text{step}\\ \hline \\ \frac{x}{20}=\frac{1-x}{10}\\ \\ 10x=20(1-x) & \text{cross-multiply}\\ \\ 10x=20-20x & \text{distribution}\\ \\ 30x=20 & \text{combine like terms}\\ \\ x=\frac{20}{30} & \text{division}\\ \\ x=\frac{2}{3} & \text{reduce the fraction}\\ \end{array} $

  Solution

  First Problem

  $\begin{array}{c|l} \text{solution} & \text{step}\\ \hline \\ \frac{x}{7}=\frac23\\ \\ 3x=14 & \text{cross-multiply}\\ \\ x=\frac{14}{3} & \text{division}\\ \\ x=4\frac{2}{3} & \text{convert to mixed number}\\ \end{array}$

  Second Problem

  $ \begin{array}{c|l} \text{solution} & \text{step}\\ \hline \\ -\frac67=\frac{x}{14}\\ \\ -84=7x & \text{cross-multiply}\\ \\ x=-12 & \text{division}\\ \end{array} $

  Third Problem

  $\begin{array}{c|l} \text{solution} & \text{step}\\ \hline \\ \frac19=\frac{x-1}{3}\\ \\ 3=9(x-1) & \text{cross-multiply}\\ \\ 3=9x-9 & \text{distribution}\\ \\ 9x=12 & \text{combine like terms}\\ \\ x=\frac{12}{9} & \text{division}\\ \\ x=\frac43 & \text{reduce fraction}\\ \\ x=1\frac13 & \text{convert to mixed number}\\ \end{array}$

  Fourth Problem

  $\begin{array}{c|l} \text{solution} & \text{step}\\ \hline \\ \frac{x}{4}=\frac{3+x}{8}\\ \\ 8x=4(3+x) & \text{cross-multiply}\\ \\ 8x=12+4x & \text{distribution}\\ \\ 4x=12 & \text{combine like terms}\\ \\ x=3 & \text{division}\\ \end{array}$

• Distinguish the linear equations from non-linear equations.

  Hints

  Here are a few clues that we may be dealing with a linear equation in disguise:

  • The variable is raised to the first power.

  • There are no variables raised to powers other than $1$.

  • There are no variables in denominators.

  • There are no variables that are rooted.

  • The equation is a proportion.

  • There is a single solution to the equation.

  Solution

  $ \begin{array}{c|c} \text{Linear equations} & \text{Non-linear Equations}\\ \hline \\ 2x+4=5 & 2x^3+5x-8=1\\ \\ 3x^1=7 & x^2=2\\\ \\ \frac{x}{2}=\frac{5}{7} & \sqrt{x+9}=12\\ \\ \frac{x-4}{5}=\frac{x}{6} & \frac{x^2}{3}=\frac56\\ \\ x=100 &\\ \\ x+2x+3x=4x+5x+6x +3 &\\ \end{array} $

• Identify the linear equation which represents the given scenario.

  Hints

  A proportion is an equation relating two ratios:

  $\frac{x}{4}=\frac23$

  When we say "the quantity", whatever follows should be put in parentheses. For example, two times the quantity, $x$, plus six is:

  $2(x+6)$

  Solution

  The ratio of Jim's height to his shadow is $\frac{5}{6}$. The ratio of the height of a flagpole to its shadow is $\frac{x}{21}$. These ratios are equal:

  $\frac56=\frac{x}{21}$

  The unknown height of a tree, $x$, is $21$ feet more than $5$ times my height of $6$ feet:

  $x=21+5(6)$

  Five times the quantity, $21$, plus $x$ is $6$:

  $5(21+x)=6$

  Five and $x$ are added, then the result is divided by $6$. Twenty-one and $x$ are added, then the result is divided by $5$. These two quantities are equal:

  $\frac{5+x}{6}=\frac{21+x}{5}$

• Review the steps for solving linear equations in disguise.

  Hints

  Here is an example of solving a linear equation in disguise, with the steps indicated:

  $ \begin{array}{c|l} \frac{x+1}{3}=\frac{5-x}{4} & \\ \\ 4(x+1)=3(5-x) & \text{cross-multiplication}\\ \\ 4x+4=15-3x & \text{distribution}\\ \\ 7x=11 & \text{combining like terms}\\ \\ x=\frac{11}{7} & \text{division}\\ \\ x=1\frac47 & \text{converting to mixed number} \end{array} $

  Solution

  $ \begin{array}{c|l} \frac{2x+5}{4}=\frac{3-x}{2} & \\ \\ 2(2x+5)=4(3-x) & \text{cross-multiplication}\\ \\ 4x+10=12-4x & \text{distribution}\\ \\ 8x=2 & \text{combining like terms}\\ \\ x=\frac28 & \text{division}\\ \\ x=\frac14 & \text{reducing fraction} \end{array} $

• Determine which equations are linear.

  Hints

  Linear equations can have variables with fractional coefficients. This is a linear equation:

  $\frac78 x + \frac19 =-\frac23 x$

  In linear equations, the variables are only raised to the power of $1$.

  This is a linear equation:

  $3x=4x^1+5$

  The constants in linear equations can feature powers other than one and the equation will still be linear. These are linear equations:

  $x+5^2=9$

  $x=\sqrt{7}$

  If the variable is raised to a power other than one, or rooted, we don't have a linear equation. These are non-linear equations:

  $4x^3+7=9$

  $3\sqrt{x}=5$

  If there is a variable in the denominator, we may not have a linear equation. This is not a linear equation:

  $\frac{x}{4}=\frac{5}{x}$

  Solution

  $x=4^2$ This equation is the same as $x=16$. That means it is linear.

  $x^2=4$ This equation features a variable squared. That means it is non-linear.

  $5x^1+\frac67=\sqrt{9}$ This equation features a variable to the first power. We can root or square any constants and the equation will still be linear.

  $5\sqrt{x}=10$ This equation features the root of a variable. This equation is non-linear.

  $\frac{x+4}{7}=\frac{3x-5}{2}$ This equation is linear.

  $\frac{7}{x+4}=\frac{3x+5}{2}$ This equation features a variable in the denominator of a fraction, and it's non-linear.

  $\frac13 x +\frac45 =\frac12 x^2$ This equation features a variable squared, so it's non-linear.

  $\frac13 x +\frac45 =\frac12 x$ This equation is linear.