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1.04 Multi-step equations

Introduction

In lesson  2.01 Properties of equality  we learned that the solution to an equation is the value (or values) that make the equation true. In this lesson, we will see that equations do not always have a unique solution, and we will continue using the properties of equality to solve increasingly complex equations.

Solutions to equations

To determine whether or not a value satisfies an equation, we need to check whether the left-hand side of the equation is equal to the right-hand side of the equation. However, an equation may not have just one answer.

Exploration

Complete the table of values by evaluating each side of equation 1:

3^{x} -5 = 3^{x}-5

x3^x-53^x-5
-1
0
1
2
3

Now, complete the table for equation 2:

9+x=x+5

x9+xx+5
-1
0
1
2
3

Finally, complete the table for equation 3:

35+5x=2x+35

x35+5x2x+35
-1
0
1
2
3
  1. What are the differences between equations 1, 2, and 3?
  2. How many solutions do you think there will be to each equation? Why?

  3. How does the structure of an equation relate to the number of solutions it has?

  4. How many solutions do you think the equation 2\left(x-8\right)=3x+4 will have? Why?

We can determine the solutions to an equation by looking at its structure. If the equation is the same on both sides, it will have infinitely many solutions. If the equation has the same variables with the same coefficients on both sides but for different constant values, there will be no solution. If the variables on both sides have different coefficients, there will be a unique solution.

Examples

Example 1

Determine how many solutions the following equations have without solving.

a

4\left(x-9\right)=x+6

Worked Solution
Create a strategy

Start by comparing both sides of the equation. We can see both sides are different and that there is an x on both sides. The coefficient on the left side is 4, and the coefficient on the right side is 1.

Apply the idea

The equation will have one unique solution.

Reflect and check

We can verify our answer by solving the equation.

\displaystyle 4\left(x-9\right)\displaystyle =\displaystyle x+6Original equation
\displaystyle 4x-36\displaystyle =\displaystyle x+6Distributive property
\displaystyle 3x-36\displaystyle =\displaystyle 6Subtraction property of equality
\displaystyle 3x\displaystyle =\displaystyle 42Addition property of equality
\displaystyle x\displaystyle =\displaystyle 14Division property of equality

There is only one solution to this equation.

b

2x-5=0.5\left(4x+10\right)

Worked Solution
Create a strategy

The first thing we notice is that both sides seem to have different values. But if we distribute the coefficient on the right side, the first term would be 2x, and the second term would be 5. The equation now has the same variables with the same coefficients on both sides, but the constant values are different.

Apply the idea

No solution

Reflect and check

We can verify our answer by solving the equation.

\displaystyle 2x-5\displaystyle =\displaystyle 0.5\left(4x+10\right)Original equation
\displaystyle 2x-5\displaystyle =\displaystyle 2x+5Distributive property
\displaystyle -5\displaystyle =\displaystyle 5Subtraction property of equality

The resulting statement is not true. This is how we know there is no solution to this equation.

Idea summary

A fully simplified equation in one variable will take one of the following three forms, corresponding to how many solutions the equation has:

  • x=a, where a is a number (a unique solution)

  • a=a, where a is a number (infinitely many solutions)

  • a=b, where a and b are different numbers (no solutions)

Creating and solving equations

There is more than one way to solve an equation, but it's important to remember that any operation must be applied to both sides of an equation. What you do to one side, you must do to the other, so the values remain equivalent. Sometimes, it is necessary to simplify an equation with the distributive property or by combining like terms before we begin solving for the variable.

In solving each equation below, the priority will be to group like terms. To do this, ask yourself what operation you must reverse.

Examples

Example 2

Solve the following equations:

a

4\left(5x+1\right)=-3\left(5x-5\right)

Worked Solution
Create a strategy

Looking at the equation, we see that the variables have different coefficients on both sides of the equation. The coefficient on the left side is 4\cdot 5=20, and the coefficient on the right side is -3\cdot 5=-15. This means it will have a unique solution.

Apply the idea
\displaystyle 4\left(5x+1\right)\displaystyle =\displaystyle -3\left(5x-5\right)Given equation
\displaystyle 20x+4\displaystyle =\displaystyle -15x+15Distributive property
\displaystyle 35x+4\displaystyle =\displaystyle 15Addition property of equality
\displaystyle 35x\displaystyle =\displaystyle 11Subtraction property of equality
\displaystyle x\displaystyle =\displaystyle \dfrac{11}{35}Division property of equality
Reflect and check

We can verify the solution by substituting it back into the equation to see if it makes the equation true.

\displaystyle 4\left(5x+1\right)\displaystyle =\displaystyle -3\left(5x-5\right)Original equation
\displaystyle 4\left[5\left(\dfrac{11}{35}\right)+1\right]\displaystyle =\displaystyle -3\left[5\left(\dfrac{11}{35}\right)-5\right]Substitute x=\dfrac{11}{35}
\displaystyle 4\left(\dfrac{11}{7}+1\right)\displaystyle =\displaystyle -3\left(\dfrac{11}{7}-5\right)Multiply 5\cdot\dfrac{11}{35}
\displaystyle 4\left(\dfrac{11}{7}+\dfrac{7}{7}\right)\displaystyle =\displaystyle -3\left(\dfrac{11}{7}-\dfrac{35}{7}\right)Create common denominators
\displaystyle 4\left(\dfrac{18}{7}\right)\displaystyle =\displaystyle -3\left(-\dfrac{24}{7}\right)Evaluate the addition
\displaystyle \dfrac{72}{7}\displaystyle =\displaystyle \dfrac{72}{7}Evaluate the multiplication

What results is a true equation, so we know x=\dfrac{11}{35} is the correct solution.

b

\dfrac{\left(x+3\right)}{2}+1.3x=\dfrac{0.8x}{4}

Worked Solution
Create a strategy

This equation has a mix of rational numbers, and the variables are in multiple terms. If we multiply both sides of the equation by the lowest common denominator, that will remove fractions and only leave us with decimals. Then, we can work on collecting all the variables.

Apply the idea

The denominators are 2 and 4, so the lowest common denominator is 4.

\displaystyle \dfrac{\left(x+3\right)}{2}+1.3x\displaystyle =\displaystyle \dfrac{0.8x}{4}Original equation
\displaystyle 4\left[\dfrac{\left(x+3\right)}{2}+1.3x\right]\displaystyle =\displaystyle 4\left(\dfrac{0.8x}{4}\right)Multiplication property of equality
\displaystyle 2\left(x+3\right)+5.2x\displaystyle =\displaystyle 0.8xDistributive property
\displaystyle 2x+6+5.2x\displaystyle =\displaystyle 0.8xDistributive property
\displaystyle 7.2x+6\displaystyle =\displaystyle 0.8xCombine like terms
\displaystyle 6\displaystyle =\displaystyle -6.4xSubtraction property of equality
\displaystyle -0.9375\displaystyle =\displaystyle xDivision property of equality
Reflect and check

There are many other ways we could have started solving this equation, but eliminating the fractions first made it much easier to solve.

Example 3

Right now, Bianca's father is 48 years older than Bianca.

Two years ago, her father was 5 times older than her.

Solve for y, Bianca's current age.

Worked Solution
Create a strategy

We want to write expressions representing Bianca's age and her father's age. Then we relate them with an equation and solve for y.

Bianca's father is currently y+48 years old.

Two years ago, Bianca was y-2 years old. Her father was y+48-2=y+46 years old.

Her father's age was five times her age at this time, which produces our equation:

y+46=5\left(y-2\right)

Apply the idea
\displaystyle y+46\displaystyle =\displaystyle 5\left(y-2\right)Original equation
\displaystyle y+46\displaystyle =\displaystyle 5y-10Distributive property
\displaystyle y+56\displaystyle =\displaystyle 5yAddition property of equality
\displaystyle 56\displaystyle =\displaystyle 4ySubtraction property of equality
\displaystyle 14\displaystyle =\displaystyle yDivision property of equality

Bianca is currently 14 years old.

Reflect and check

Let's check by referring back to the information given in the problem. Bianca's father is 48 years older, so he is 14+48=62 years old. Two years ago, he was 5 times older. Two years ago, Bianca would have been 12, and her father would have been 60. Is this 5 times Bianca's age? Yes, this is correct.

Idea summary

We can simplify an equation using the distributive property or the multiplication property of equality to eliminate fractions or decimals. After simplifying, we can continue using properties of equality to solve for the unknown.

Outcomes

A.CED.A.1

Create equations and inequalities in one variable and use them to solve problems.

A.REI.B.3

Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters.

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