Chapter 8.8, Problem 58HP

To determine

Explain how to determine if an equation has no solution, one solution or if all numbers are solutions. Use examples with you explanation.

Answer to Problem 58HP

Explanation of Solution

Given:

Explain how to determine if an equation has no solution, one solution or if all numbers are solutions. Use examples with you explanation.

Concept Used:

If an equation is x = a, this means the equation is true only when the variable assumes the value a. The equation has only one solution exists.

If an equation a = a, this means the equation is true for any value of the variable, has infinitely many solution.

If a = b, this means there is no value of the variable that will make a equal to b , has no solution.

One unique Solution:

Example: $5x-2+3x=3\left(x+4\right)-1$

To solve, we will collect the like terms on the left side of the equal sign and distribute the 3 on the right side of equal sign

  $5x-2+3x=3\left(x+4\right)-1$ is equivalent to $8x-2=3x+12-1$

We can write $8x-2=3x+12-1\Rightarrow 8x-2=3x+11$

Now solve for x:

  $\begin{array}{l}8x-2=3x+11\\ 8x-3x-2=11[\text{Subtract 3x from both sides ]}\\ \text{5x }-2=11\\ 5x-2+2=11+2[\text{add}\text{2}\text{to}\text{both}\text{sides}]\\ \frac{5x}{5}=\frac{13}{5}[\text{Divide each side by 5 ] }\\ x=\frac{13}{5}\end{array}$

This equation happens to have a unique answer, which is $x=\frac{13}{5}$

Explanation for No Solution:

When finding how many solutions an equation has, you need to look at the constants and coefficients.

The coefficients are the numbers alongside the variables.

The constants are the numbers alone with no variables.

If the coefficients of variable are the same on both sides are equal, but the constants are different, then no solutions will occur.

If the equation ends with a false statement (example: a = b) then you know that there`s no solution.

Example:

  $9x+4=3(3x+4)$

Use distributive property on the right side first.

  $\begin{array}{l}9x+4=3(3x+4)\\ 9x+4=9x+12\\ 9x-9x+4=9x-9x+12[\text{Subtract 9x from both sides ]}\\ 4\ne 12\end{array}$

No Solution.

Therefore we can conclude that there is no solution to this equation. Whatever number we put in for the variable x, it'll never give us a true statement.

Explanation for Infinitely many solutions:

If the equation ends with a true statement (example: a = a) then you know that there`s infinitely many solutions or all real numbers.

Example:

  $\begin{array}{l}x-10+x=8+2x-18\\ 2x-10=2x-10[\text{Simplify]}\\ 2x-2x-10=2x-2x-10[\text{Subtract 9x from both sides ]}\\ -10=-10\end{array}$

Once again, the x terms cancel out, but we are left with a true statement. -10 does equal -10, and will no matter what we put in for x.

Therefore we can conclude that there is infinitely many solutions. Whatever number we put in for the variable x, it'll always give you a true statement.

If we look carefully at the equation, $2x-10=2x-10$ , we will see that for any x you substitute on both sides of the equation the results will be the same so the solution to this equation is x is real, that is, any number x will satisfy this equation.