Solving One-Step Equations

Solving One-Step Equations exercise

• Write an equation that best represents the number of missing baby lions.

  Hints

  An equation is like a scale: both sides of the scale have to have the same value in order to be in balance.

  The $x$ in the equation represents the number of missing baby lions.

  The total number of baby lions in the zoo is $7$, which represents one side of the equation.

  The other side of the equation also represents the total number of baby lions in the zoo, but this time we have the unknown number of missing baby lions, $x$, and the number of baby lions that are still in the cage, $2$.

  Solution

  To write this real world problem as a math equation, it helps if we think of the equation as a scale that must be balanced.

  We know the total number of baby lions in the zoo: $7$.

  What we want to know is the number of missing baby lions, so we assign this unknown quantity to the variable $x$. We know that the total number of baby lions is the number of missing baby lions plus the number of baby lions that are still in the cage. This can be represented by the expression $x + 2$.

  This expression must be equal to the total number of baby lions, $7$. Writing out the full equation, we get $x + 2 = 7$, where $x$ represents the number of missing baby lions.

• Describe how to solve a one-step equation.

  Hints

  Let's take a look at an example:

  • $x+2=7$

  So how do we get rid of the $+2$ on the left side of the equation in order to isolate the $x$? Can it be done by adding $2$ to both sides of the equation?

  Let's see:

  $\begin{array}{rcr} x+2 &=&7\\ \color{#669900}{~~+2} & ~~ &\color{#669900}{+2}\\ x+4 &=&9\\ \end{array}$

  Hmm, as we can see, the equation is more difficult than before and the $x$ is still not isolated. Can you guess what we should have done?

  So, if we want to isolate $x$ in the equation $x+2=7$ it doesn't really make sense to add $2$ on both sides of the equation. Instead, subtracting $2$ on both sides should do the trick. Let's see:

  $\begin{array}{lcr} x+2 & =& 7\\ \color{#669900}{~~-2} & ~~ & \color{#669900}{-2}\\ x & = & 5 \end{array}$

  Great. as you can see, the $x$ is isolated on the left side.

  To get rid of a term on one side of an equation, you have to use the opposite operation.

  Solution

  When you solve an equation, you have to isolate the variable on one side of the equation. To do so, you must use the opposite operation of the term you want to get rid of.

  Remember, an equation is like a balanced scale, so whatever you do to one side, you have to do the other.

  The opposite operation of addition is subtraction, and vice versa. Let's take a look at two examples:

  $\begin{array}{lcr} x+2 & = & 7\\ \color{#669900}{~~-2} & ~~ & \color{#669900}{-2}\\ x & = & 5 \end{array}$

  Since we have a $-4$, we have to do the opposite operation, $+$, to both sides of the equation:

  $\begin{array}{lcr} x-4& = & 8 \\ \color{#669900}{~~+4} & ~~ & \color{#669900}{+4}\\ x & = & 12 \end{array}$

  The opposite operation of multiplication is division, and vice versa. Let's take a look at two examples:

  $\begin{array}{lcr} x \times 2 & = & 10\\ ~~ \color{#669900}{\div 2} & ~~ & \color{#669900}{\div 2}\\ x & =&5 \end{array}$

  Now we perform the opposite operation of $\times$, $\div$:

  $\begin{array}{lcr} x\div 2 & = & 3\\ ~~ \color{#669900}{\times 2} & ~~ & \color{#669900}{\times 2}\\ x & = & 6 \end{array}$

• Find out how many giraffes belong to the giraffe family.

  Hints

  The number of giraffes is a natural number not a fraction. You can't have 1/2 of a giraffe!

  The opposite operation of addition is subtraction, and vice versa.

  The opposite operation of multiplication is division, and vice versa.

  To isolate the variable, you must carry out the opposite operation of the term you want to move on both sides of the equation.

  Solution

  We know the number of missing giraffes, $4$. We also know that the number of missing giraffes is a third of the total number, which is unknown. We represent the unknown number with the variable $x$.

  One third of $x$ can be expressed mathematically as $x\div 3$. To find the total number of giraffes, we set $x\div 3$ equal to $4$ and solve for $x$.

  $x\div 3=4$.

  To do this, we have to do the opposite of $x\div3$ on both sides of the equation. By multiplying by $3$ on both sides of the equation, we get our solution: $x=12$.

  There are $12$ giraffes in total.

• Find the solution for the following equations.

  Hints

  The opposite operation of addition is subtraction, and vice versa.

  The opposite operation of multiplication is division, and vice versa.

  To check your work, you can plug the solution for $x$ into the original equation. If your answer checks out, you will have a true math statement.

  Remember, to isolate a variable, use the opposite operation on both sides of the equation to remove terms.

  Solution

  We can solve equations by isolating the variable $x$. Then we solve for $x$ using opposite operations.

  Let's try it with several examples.

  The opposite operation of addition is subtraction, and vice versa:

  $\begin{array}{lcr} x+3 & = & 12\\ ~~\color{#669900}{-3} & ~~ & \color{#669900}{-3}\\ x & = & 9 \end{array}$

  Let's do the same with subtraction:

  $\begin{array}{lcr} x-7 & = & 5\\ ~~\color{#669900}{+7} & ~~ & \color{#669900}{+7}\\ x & = & 12 \end{array}$

  The opposite operation of division is multiplication, and vice versa:

  $\begin{array}{lcr} x \times 2 & =& 16\\ ~~ \color{#669900}{\div 2}& ~~ & \color{#669900}{\div 2}\\ x & = & 8 \end{array}$

  Let's do the same with division:

  $\begin{array}{lcr} x \div 5 & = & 3\\ ~~\color{#669900}{\times 5} & ~~ & \color{#669900}{\times 5} \\ x & = & 15 \end{array}$

• Determine the correct steps needed to solve an equation.

  Hints

  How does a scale work? Both sides of the scale must have equal weights in order to be in balance.

  Whatever you do on one side of an equation, think of what you have to do the other side of the equation in order to keep it in balance.

  Solution

  An equation is defined by two expressions joined by an equal sign.

  The equal sign is like the middle of a scale. The equation corresponds to a balanced scale. If you take away something on one side of the scale, you have to do the same on the other side as well. Otherwise, the scale isn't in balance any longer.

  The same concept applies to equations:

  $\begin{array}{lcr} x+2 & = & 7\\ ~~\color{#669900}{-2} & ~~ & \color{#669900}{-2}\\ x & = & 5 \end{array}$

  The operations we have learned so far are:

  • You can add or subtract a term on both sides of an equation.
  • You can multiply both sides of an equation by the same, non-zero term.
  • You can divide both sides of the equation by the same, non-zero term.

• Solve the following equations.

  Hints

  Each solution is a multiple of $10$.

  Carry out the opposite operation

  • $+\longleftrightarrow -$
  • $\times \longleftrightarrow \div$

  The greatest solution is $x=50$.

  Solution

  To solve equations, we have to isolate the variable $x$ using opposite operations.

  • The opposite operation of addition is subtraction, and vice versa.
  • The opposite operation of multiplication is division, and vice versa.

  The equations in the correct order of their solutions from least to great are:

  $\begin{array}{lcr} x \times 3 & = & 60\\ ~~ \color{#669900}{\div 3} & ~~ & \color{#669900}{\div 3}\\ x & = & 20\\ ~&~&~\\ ~&~&~\\ x+15 & = & 45\\ ~~\color{#669900}{-15} & ~~ & \color{#669900}{-15}\\ x & = & 30\\ ~&~&~\\ ~&~&~\\ x \div 5 & = & 8\\ ~~ \color{#669900}{\times 5} & ~~ & \color{#669900}{\times 5}\\ x & = & 40\\ ~&~&~\\ ~&~&~\\ \end{array}$

  The greatest $x$ value is:

  $\begin{array}{lcr} x-10 & = & 40\\ ~~ \color{#669900}{+10} & ~~ & \color{#669900}{+10}\\ x & = & 50 \end{array}$