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Mathematics LibreTexts

2.3E: Exercises

  • Page ID
    30478
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    Practice Makes Perfect

    Solve Equations with Constants on Both Sides

    In the following exercises, solve the following equations with constants on both sides.

    Exercise \(\PageIndex{1}\)

    \(9 x-3=60\)

    Exercise \(\PageIndex{2}\)

    \(12 x-8=64\)

    Answer

    \(x=6\)

    Exercise \(\PageIndex{3}\)

    \(14 w+5=117\)

    Exercise \(\PageIndex{4}\)

    \(15 y+7=97\)

    Answer

    \(y=6\)

    Exercise \(\PageIndex{5}\)

    \(2 a+8=-28\)

    Exercise \(\PageIndex{6}\)

    \(3 m+9=-15\)

    Answer

    \(m=-8\)

    Exercise \(\PageIndex{7}\)

    \(-62=8 n-6\)

    Exercise \(\PageIndex{8}\)

    \(-77=9 b-5\)

    Answer

    \(b=-8\)

    Exercise \(\PageIndex{9}\)

    \(35=-13 y+9\)

    Exercise \(\PageIndex{10}\)

    \(60=-21 x-24\)

    Answer

    \(x=-4\)

    Exercise \(\PageIndex{11}\)

    \(-12 p-9=9\)

    Exercise \(\PageIndex{12}\)

    \(-14 q-2=16\)

    Answer

    \(q=-\frac{9}{7}\)

    Solve Equations with Variables on Both Sides

    In the following exercises, solve the following equations with variables on both sides.

    Exercise \(\PageIndex{13}\)

    \(19 z=18 z-7\)

    Exercise \(\PageIndex{14}\)

    \(21 k=20 k-11\)

    Answer

    \(k=-11\)

    Exercise \(\PageIndex{15}\)

    \(9 x+36=15 x\)

    Exercise \(\PageIndex{16}\)

    \(8 x+27=11 x\)

    Answer

    \(x=9\)

    Exercise \(\PageIndex{17}\)

    \(c=-3 c-20\)

    Exercise \(\PageIndex{18}\)

    \(b=-4 b-15\)

    Answer

    \(b=-3\)

    Exercise \(\PageIndex{19}\)

    \(9 q=44-2 q\)

    Exercise \(\PageIndex{20}\)

    \(5 z=39-8 z\)

    Answer

    \(z=3\)

    Exercise \(\PageIndex{21}\)

    \(6 y+\frac{1}{2}=5 y\)

    Exercise \(\PageIndex{22}\)

    \(4 x+\frac{3}{4}=3 x\)

    Answer

    \(x=-\frac{3}{4}\)

    Exercise \(\PageIndex{23}\)

    \(-18 a-8=-22 a\)

    Exercise \(\PageIndex{24}\)

    \(-11 r-8=-7 r\)

    Answer

    \(r=-2\)

    Solve Equations with Variables and Constants on Both Sides

    In the following exercises, solve the following equations with variables and constants on both sides.

    Exercise \(\PageIndex{25}\)

    \(8 x-15=7 x+3\)

    Exercise \(\PageIndex{26}\)

    \(6 x-17=5 x+2\)

    Answer

    \(x=19\)

    Exercise \(\PageIndex{27}\)

    \(26+13 d=14 d+11\)

    Exercise \(\PageIndex{28}\)

    \(21+18 f=19 f+14\)

    Answer

    \(f=7\)

    Exercise \(\PageIndex{29}\)

    \(2 p-1=4 p-33\)

    Exercise \(\PageIndex{30}\)

    \(12 q-5=9 q-20\)

    Answer

    \(q=-5\)

    Exercise \(\PageIndex{31}\)

    \(4 a+5=-a-40\)

    Exercise \(\PageIndex{32}\)

    \(8 c+7=-3 c-37\)

    Answer

    \(c=-4\)

    Exercise \(\PageIndex{33}\)

    \(5 y-30=-5 y+30\)

    Exercise \(\PageIndex{34}\)

    \(7 x-17=-8 x+13\)

    Answer

    \(x=2\)

    Exercise \(\PageIndex{35}\)

    \(7 s+12=5+4 s\)

    Exercise \(\PageIndex{36}\)

    \(9 p+14=6+4 p\)

    Answer

    \(p=-\frac{8}{5}\)

    Exercise \(\PageIndex{37}\)

    \(2 z-6=23-z\)

    Exercise \(\PageIndex{38}\)

    \(3 y-4=12-y\)

    Answer

    \(y=4\)

    Exercise \(\PageIndex{39}\)

    \(\frac{5}{3} c-3=\frac{2}{3} c-16\)

    Exercise \(\PageIndex{40}\)

    \(\frac{7}{4} m-7=\frac{3}{4} m-13\)

    Answer

    \(m=-6\)

    Exercise \(\PageIndex{41}\)

    \(8-\frac{2}{5} q=\frac{3}{5} q+6\)

    Exercise \(\PageIndex{42}\)

    \(11-\frac{1}{5} a=\frac{4}{5} a+4\)

    Answer

    \(a=7\)

    Exercise \(\PageIndex{43}\)

    \(\frac{4}{3} n+9=\frac{1}{3} n-9\)

    Exercise \(\PageIndex{44}\)

    \(\frac{5}{4} a+15=\frac{3}{4} a-5\)

    Answer

    \(a=-40\)

    Exercise \(\PageIndex{45}\)

    \(\frac{1}{4} y+7=\frac{3}{4} y-3\)

    Exercise \(\PageIndex{46}\)

    \(\frac{3}{5} p+2=\frac{4}{5} p-1\)

    Answer

    \(p=15\)

    Exercise \(\PageIndex{47}\)

    \(14 n+8.25=9 n+19.60\)

    Exercise \(\PageIndex{48}\)

    \(13 z+6.45=8 z+23.75\)

    Answer

    \(z=3.46\)

    Exercise \(\PageIndex{49}\)

    \(2.4 w-100=0.8 w+28\)

    Exercise \(\PageIndex{50}\)

    \(2.7 w-80=1.2 w+10\)

    Answer

    \(w=60\)

    Exercise \(\PageIndex{51}\)

    \(5.6 r+13.1=3.5 r+57.2\)

    Exercise \(\PageIndex{52}\)

    \(6.6 x-18.9=3.4 x+54.7\)

    Answer

    \(x=23\)

    Everyday Math

    Exercise \(\PageIndex{53}\)

    Concert tickets At a school concert the total value of tickets sold was $1506. Student tickets sold for $6 and adult tickets sold for $9. The number of adult tickets sold was 5 less than 3 times the number of student tickets. Find the number of student tickets sold, s, by solving the equation 6s+27s−45=1506. Add exercises text here.

    Exercise \(\PageIndex{54}\)

    Making a fence Jovani has 150 feet of fencing to make a rectangular garden in his backyard. He wants the length to be 15
    feet more than the width. Find the width, w, by solving the equation \(150=2 w+30+2 w\).

    Answer

    30 feet

    Writing Exercises

    Exercise \(\PageIndex{55}\)

    Solve the equation \(\frac{6}{5} y-8=\frac{1}{5} y+7\) explaining all the steps of your solution as in the examples in this section.

    Exercise \(\PageIndex{56}\)

    Solve the equation \(10 x+14=-2 x+38\) explaining all the steps of your solution as in this section.

    Answer

    \(x=2\) Justifications will vary.

    Exercise \(\PageIndex{57}\)

    When solving an equation with variables on both sides, why is it usually better to choose the side with the larger coefficient
    of \(x\) to be the "variable" side?

    Exercise \(\PageIndex{58}\)

    Is \(x=-2\) a solution to the equation \(5-2 x=-4 x+1 ?\) How do you know?

    Answer

    Yes. Justifications will vary.

    Self Check

    ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

    This is a table that has four rows and four columns. In the first row, which is a header row, the cells read from left to right: “I can...,” “Confidently,” “With some help,” and “No-I don’t get it!” The first column below “I can...” reads: “solve an equation with constants on both sides,” “solve an equation with variables on both sides,” and “solve an equation with variables and constants on both sides. ” The rest of the cells are blank.

    ⓑ What does this checklist tell you about your mastery of this section? What steps will you take to improve?