2.02 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
| x | 3^x-5 | 3^x-5 |
|---|---|---|
| -1 | ||
| 0 | ||
| 1 | ||
| 2 | ||
| 3 |
Now, complete the table for equation 2:
9+x=x+5
| x | 9+x | x+5 |
|---|---|---|
| -1 | ||
| 0 | ||
| 1 | ||
| 2 | ||
| 3 |
Finally, complete the table for equation 3:
35+5x=2x+35
| x | 35+5x | 2x+35 |
|---|---|---|
| -1 | ||
| 0 | ||
| 1 | ||
| 2 | ||
| 3 |
- What are the differences between equations 1, 2, and 3?
How many solutions do you think there will be to each equation? Why?
-
How does the structure of an equation relate to the number of solutions it has?
- 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.
4\left(x-9\right)=x+6
2x-5=0.5\left(4x+10\right)
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:
4\left(5x+1\right)=-3\left(5x-5\right)
\dfrac{\left(x+3\right)}{2}+1.3x=\dfrac{0.8x}{4}
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.
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.