"

Solve the following equations for the unknown variable. #1-4 are one step equations because you only have undo one operation to isolate the variable.

  1. [latex]x\;-\;7\;=\;10[/latex]
  2. [latex]5\;+\;y\;=\;8[/latex]
  3. [latex]7x\;=\;-21[/latex]
  4. [latex]-3y\;=\;42[/latex]
    1. Are [latex]-3y\;=\;42[/latex] and [latex]-3\;+\;y\;=\;42[/latex] the same thing? Why or why not?
    2. How would you solve [latex]-3\;+\;y\;=\;42[/latex]?
    3. In your words, explain how part (b) was different from solving #4.
  6. Spot the error! Explain in your own words the mistake made and correct it.
    [latex]8x\;=\;4[/latex]
    [latex]\frac{8x}4=\frac44[/latex]
    [latex]x\;=\;2[/latex]
  7. Spot the error! Explain in your own words the mistake made and correct it.
    [latex]8x\;=\;4[/latex]
    [latex]\frac{8x}{8x}=\frac4{8x}[/latex]
    [latex]x\;=\;\frac12[/latex]

#8-14 are multistep equations because you have undo more than one operation to isolate the variable.

  1. [latex]2x\;-\;5\;=\;7[/latex]
  2. [latex]2\;+\;3y\;=\;17[/latex]
  3. [latex]3\;-\;3x\;=\;9[/latex]
  4. [latex]1\;-\;y\;=\;-10[/latex]
  5. [latex]8\;-\;4y\;=\;18[/latex]
  6. [latex]-2y\;-\;4\;=\;-1[/latex]
  7. Solve [latex]5\;-\;6y\;=\;-7[/latex] two different ways.
    1. Start by subtracting 5 from both sides.
    2. Start by adding 6y to both sides.
    3. Which way do you prefer and why?
  8. Spot the errors! Explain in your own words the mistakes made and correct them.
    [latex]3\;-\;4x\;=\;-9[/latex]
    [latex]3\;-\;4x\;+\;9\;=\;-9\;+\;9[/latex]
    [latex]12\;-\;4x[/latex]
    [latex]\frac{12}4-\frac{4x}4[/latex]
    [latex]x\;=\;3[/latex]

When you see a number held together to an expression by parentheses such as [latex]4(x\;-\;5)[/latex], what operation is implied (addition, subtraction, multiplication, or division)?

The distributive property of multiplication over addition

[latex]a(b\;+\;c)\;=\;a\;\cdot\;b\;+\;a\;\cdot\;c[/latex]

Try it! Simplify the following expressions by distributing.

  1. [latex]4(x\;-\;5)[/latex]
  2. [latex]4\left(\frac x4-\frac54\right)[/latex]
  3. [latex]4\left(\frac12-\frac52y\right)[/latex]
  4. [latex]12\left(\frac x4-\frac53\right)[/latex]

When you see an expression held together by a fraction bar such as [latex]\frac{x-6}3[/latex], what operation is implied (addition, subtraction, multiplication, or division)?

And how can you undo this operation?

Try it! Clear the following equations of the fractions and then solve.

  1. [latex]\frac{x-6}3=\frac13[/latex]
  2. [latex]\frac x4-\frac54=\frac14[/latex] What if the denominators are not all the same number?
  3. [latex]\frac x2-\frac34=3[/latex]
  4. [latex]\frac x4-\frac53=1[/latex]
  5. [latex]\frac12-\frac23y=4[/latex]
  6. [latex]\frac12-\frac{5y}2=\frac34[/latex] What if the variable shows up on both sides of the equation?
  7. [latex]8x\;+\;9\;=\;7x\;-\;3[/latex]
  8. [latex]6(3\;+\;x)\;=\;5(x\;+\;1)[/latex]
  9. [latex]\frac3{1+x}=2[/latex]
  10. [latex]\frac{28+x}{7+x}=2[/latex]

License

College Algebra for Non-STEM Majors Copyright © by Amy Collins Montalbano. All Rights Reserved.

Share This Book