One-Step Equations with Multiplication and Division

One-Step Equations with Multiplication and Division exercise

• Recall how to solve one-step equations.

  Hints

  Henry knows that he needs $6$ pounds of food per day and that he can carry $36$ pounds. How many days will Henry have food?

  Let $d$ represents the number of days, the equation is then $6d=36$.

  If Henry walks at an average speed of $12$ miles per day, for $3$ days, how many total miles can Henry walk?

  Let $m$ be the total number of miles. The equation is then $\frac{m}{3}=12$.

  To solve $6d=36$, divide both sides of the equation by $6$. We then have that $d=6$.

  To solve $\frac{m}{3}=12$, multiply both sides of the equation by 3. We then have that $m=36$.

  Solution

  Henry wants to calculate how many days his food will last him. He knows that he will need $5$ pounds of food per day and that he can carry $20$ pounds in his backpack. To figure this out, let's set up an equation.

  • First, we need to identify the unknown. Since we don't know the number of days his food will last him, we will let the variable $d$ represent the number of days.
  • Since Henry knows that he needs $5$ pounds per day, we can translate this phrase into the expression $5d$. We are told he can carry a total of $20$ pounds in his backpack. Therefore, the equation is, $5d=20$.
  • To solve for the unknown we will need to isolate $d$.
  • Since $d$ is being multiplied by $5$, we need to use opposite operations and divide by $5$ on both sides to keep the equation balanced.
  • Doing so, we have $5d\div 5=20\div 5$.
  • The $5$'s cancel out on the left side of the equation. We then find that $d=4$.

  Great job! You helped Henry use a one-step equation to figure out that he has $4$ days worth of food.

  Now Henry needs help figuring out how many miles he can walk in $4$ days if his average speed is $11$ miles per day.

  This problem can be represented as the math equation, $\frac{\text{miles}}{\text{days}}=\text{miles per day}$.

  • First, we need to identify the unknown. Since we don't know the number of miles Henry can walk in $4$ days, we will let the variable $m$ represent the total number of miles.
  • Now we can fill in the equation, $\frac{\text{miles}}{\text{days}}=\text{miles per day}$, by substituting $m$ for miles, $4$ for days, and $11$ for miles per day. The equation we get is then $\frac{m}{4}=11$.
  • To solve for the unknown we will need to isolate $m$.
  • Since $m$ is being divided by $4$, we will use opposite operations and multiply by $4$ on both sides to keep the equation balanced.
  • Doing so, we have $\frac{m}{4}\cdot 4=11\cdot 4$.
  • The $4$'s cancel out on the left side of the equation. Therefore, $m=44$.

• Identify which operation needs to be used to isolate the variable.

  Hints

  To isolate a variable, use the opposite operation.

  For example, if a number is being added to the variable $m$, then use subtraction get isolate $m$.

  $x+2=3$

  • Since $2$ is being added to $x$, use subtraction to isolate $x$.

  $x-2=3$

  • Since $2$ is being subtracted from $x$, use addition to isolate $x$.

  $2x=3$

  • Since $x$ is being multiplied by $2$, use division to isolate $x$.

  $\frac{x}{2}=3$

  • Since $x$ is being divided by $2$, use multiplication to isolate $x$.

  Solution

  Use the opposite operations to isolate the variable in the equation.

  This is also known as the additive inverse for addition and subtraction problems, and the multiplicative inverse for multiplication and division problems.

  $y-32=6$

  • Since $32$ is being subtracted from $y$, use addition to isolate $y$.

  $-74+p=110$

  • Since $74$ is being subtracted from $p$, use addition to isolate $p$.

  $r+5=12$

  • Since $5$ is being added to $r$, use subtraction to isolate $r$.

  $1+m=17$

  • Since $1$ is being added to $m$, use subtraction to isolate $m$.

  $\frac{c}{4}=18$

  • Since $c$ is being divided by $4$, use multiplication to isolate $c$.

  $\frac{m}{5}=12$

  • Since $m$ is being divided by $5$, use multiplication to isolate $m$.

  $6d=30$

  • Since $d$ is being multiplied by $6$, use division to isolate $d$.

  $4x=240$

  • Since $x$ is being multiplied by $4$, use division to isolate $x$.

• Write and solve one-step equations.

  Hints

  Solve each equation by thinking about which opposite operation must be applied to the equation to get the variable by itself.

  In the equation $5x=35$, since $x$ is being multiplied by $5$, the opposite operation is to divide both sides by $5$ so that $x=7$.

  In the equation $x-6=12$, since $6$ is being subtracted from $x$, the opposite operation is to add $6$ to both sides so that $x=18$.

  Solution

  The number of days Henry's water supply will last if he needs $3$ bottles per day and he can carry $24$ in his backpack.

  • Let $x$ be the number of days.
  • The equation is $3x=24$.
  • The opposite operation is to divide both sides by $3$.
  • $x=8$

  The number of miles Henry needs to walk on day $5$ of his backpacking trip if he already walked $48$ miles. His total number of miles is $60$.

  • Let $x$ be the number of miles.
  • The equation is $x+48=60$.
  • The opposite operation is to subtract $48$ from both sides.
  • $x=12$

  The number of hot dogs Henry has on day two of his backpacking trip if he ate $5$ on the first day. He is traveling with $30$ hot dogs.

  • Let $x$ be the number of hot dogs.
  • The equation is $x-5=30$.
  • The opposite operation is to add $5$ to both sides.
  • $x=35$

  The total number of hours Henry spends hiking if he hikes $6$ days at a rate of $7$ hours per day.

  • Let $x$ be the number of hours.
  • The equation is $\frac{x}{6}=7$.
  • The opposite operation is to multiply each side by $6$.
  • $x=42$

• Calculate the solution for each one-step equation.

  Hints

  • $m+6$ is the same as $6+m$ because of the commutative property of addition.

  • $m6$ is the same as $6m$ because of the commutative property of multiplication.

  To solve a one-step equation, use the opposite operation to isolate the variable.

  $m+6=12$

  • Opposite operation is to subtract $6$.

  $m-6=12$

  • Opposite operation is to add $6$.

  $m\cdot 6=12$

  • Opposite operation is to divide $6$.

  $\frac{m}{6}=12$

  • Opposite operation is to multiply $6$.

  Solution

  1. $4m=-12$

  • The opposite operation is to divide by $4$ on both sides.
  • $\frac{4m}{4}=\frac{-12}{4}$
  • $m=-3$

  2. $6+m=12$

  • The equation can also be written as $m+6=12$ because of the commutative property of addition.
  • The opposite operation is to subtract $6$ from both sides.
  • $m+6-6=12-6$
  • $m=6$

  3. $\frac{m}{-2}=4$

  • The opposite operation is to multiply $-2$ on both sides.
  • $\frac{m}{-2}\cdot -2=4\cdot -2$

  4. $m-9=2$

  • The opposite operation is to add $9$ to both sides.
  • $m-9+9=2+9$
  • $m=11$

  5. $m\cdot 7=14$

  • The equation can also be written as $7\cdot m=14$ because of the commutative property of multiplication.
  • The opposite operation is to divide by $7$ on both sides.
  • $\frac{7\cdot m}{7}=\frac{14}{7}$
  • $m=2$

  6. $m+10=15$

  • The opposite operation is to subtract $10$ from both sides.
  • $m+10-10=15-10$
  • $m=5$

• Determine which statements are true about one-step equations.

  Hints

  $k+3=15$

  • Since $1$ is being added to $k$, we need to use the opposite operation and subtract $1$ from both sides of the equation to isolate the variable $k$.
  • Using the opposite operation is the same as using the additive inverse.

  $3k=15$

  • Since $k$ is being multiplied by $3$, we need to multiply by the multiplicative inverse of $3$, which is $\frac {1}{3}$, to isolate the variable $k$.
  • Using the multiplicative inverse is the same as using the opposite operation.

  $\frac{k}{3}=15$

  • Since $k$ is being divided by $3$, we need to multiply both sides by $3$ to isolate the variable $k$.

  Solution

  True Statements

  • For the equation $d+7=10$, to isolate the variable $d$ we must add the additive inverse of $7$, which is $-7$, to both sides of the equation.
  • For the equation $4d=16$, since $d$ is being multiplied by 4, we must use the opposite operation and divide both sides of the equation by $4$ to isolate the variable $d$.
  • For the equation $\frac{x}{2}=24$, to isolate the variable $x$ we must multiply both sides of the equation by the multiplicative inverse of $\frac{1}{2}$, which is $2$.
  • For the equation $c-8=9$, since $8$ is being subtracted from $c$, we must use the opposite operation and add $8$ to both sides of the equation in order to isolate the variable $c$.

  $~$

  False Statements

  For the equation $m+2=15$, to isolate the variable $m$, we must add the additive inverse of $15$, which is $-15$, to both sides of the equation.

  • The correct statement is: For the equation $m+2=15$, to isolate the variable $m$, we must add the additive inverse of $2$, which is $-2$, to both sides of the equation.

  For the equation $3y=21$, to isolate the variable $y$ we must multiply by the multiplicative inverse of $21$, which is $\frac{1}{21}$, to both sides of the equation.

  • The correct statement is: For the equation $3y=21$, to isolate the variable $y$ we must multiply by the multiplicative inverse of $3$, which is $\frac{1}{3}$, to both sides of the equation.

• Solve the one-step and two-step equations.

  Hints

  To solve one-step and two-step equations, think of the opposite operations that are needed to isolate the variable.

  One-Step Equations:

  • If $-2m=8$, divide both sides by $-2$ so that $m=-4$.
  • If $\frac{m}{-2}=8$, multiply both sides by $-2$ so that $m=-16$.
  • If $m-2=8$, add $2$ to both sides so that $m=10$.
  • If $m+2=8$, subtract $2$ from both sides so that $m=6$.

  Two-step equations are very similar to one-step equations. The only difference is that you have to think about two opposite operations to get the variable alone.

  • $2m-5=11$
  • The first opposite operation is to add $5$ to both sides, so that $2m=16$.
  • The second opposite operation is to divide both sides by $2$ to get $m=16$.

  Solution

  One-Step Equations:

  1. $-2m=24$

  • The opposite operation is to divide both sides by $-2$ so that $m=-12$.

  2. $\frac{m}{4}=7$

  • The opposite operation is to multiply both sides by $4$ so that $m=28$.

  Two-Step Equations:

  • Two-step equations are very similar to one-step equations. The only difference is that you have to think about the two opposite operations to get the variable alone.

  3. $2m+5=19$

  • The first opposite operation is to subtract $5$ from both sides to get $2m=14$
  • The second opposite operation is to divide both sides by $2$ to get $m=7$.

  4. $-3+5m=7$

  • The first opposite operation is to add $3$ to both sides to get $5m=10$.
  • The second opposite operation is to divide both sides by $5$ to get $m=2$.

  5. $2m-5=3$

  • The first opposite operation is to add $5$ to both sides to get $2m=8$.
  • The second opposite operation is to divide both sides by $2$ to get $m=4$.