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MathAlgebra

Lesson 3.4 Solving Complex 1-Variable Equations Answer Key

Solve the complete set of 24 Lesson 3.4 multi-step equations using distribution, like-term collection, exact fractions, and substitution checks.

Question

Lesson 3.4: Solving Complex 1-Variable EquationsSolve each equation.1a. 3x + 4 + 2x + 5 = 341b. 2(x + 1) + 4 = 122a. 1/2(x + 8) - 15 = -32b. 2x - 5 + 3x + 8 = 183a. -185 = -3r - 4(-5r + 8)3b. -5t - 2(5t + 10) = 1004a. -4b - 4(-6b - 8) = 1724b. -3p + 2(5p - 12) = -735a. -3f + 3(-3f + 5) = -815b. -43 = -5c + 4(2c + 7)6a. -5s + 3(5s + 2) = 1266b. 4d + 2(4d + 7) = -1067a. 103 = -2u + 3(-3u + 5)7b. -2n + 2(3n + 14) = -208a. -11 = 5y + 4(-y - 4)8b. -5a - 2(-7a - 10) = 1289a. 1/2(c + 5) - 10 = -49b. -4f + 1/2(4f - 5) = -1910a. 2(v + 4) + 6 = 2410b. -9 = 6h + 3(-h - 3)11a. -6p - 8(4p + 8) = 9811b. 7c + 3(3c + 5) = -10312a. -4s + 2(4s + 1) = 12512b. -3n + 3(4n + 15) = -21

Answer key

The reliable pattern is: distribute first, combine like terms, isolate the variable, and substitute the result into the original equation.

ProblemSimplified equationAnswer
1a5x+9=345x+9=34x=5x=5
1b2x+6=122x+6=12x=3x=3
2ax+82=12\frac{x+8}{2}=12x=16x=16
2b5x+3=185x+3=18x=3x=3
3a17r−32=−18517r-32=-185r=−9r=-9
3b−15t−20=100-15t-20=100t=−8t=-8
4a20b+32=17220b+32=172b=7b=7
4b7p−24=−737p-24=-73p=−7p=-7
5a−12f+15=−81-12f+15=-81f=8f=8
5b3c+28=−433c+28=-43c=−713c=-\frac{71}{3}
6a10s+6=12610s+6=126s=12s=12
6b12d+14=−10612d+14=-106d=−10d=-10
7a−11u+15=103-11u+15=103u=−8u=-8
7b4n+28=−204n+28=-20n=−12n=-12
8ay−16=−11y-16=-11y=5y=5
8b9a+20=1289a+20=128a=12a=12
9ac+52=6\frac{c+5}{2}=6c=7c=7
9b−2f−52=−19-2f-\frac52=-19f=334f=\frac{33}{4}
10a2v+14=242v+14=24v=5v=5
10b3h−9=−93h-9=-9h=0h=0
11a−38p−64=98-38p-64=98p=−8119p=-\frac{81}{19}
11b16c+15=−10316c+15=-103c=−598c=-\frac{59}{8}
12a4s+2=1254s+2=125s=1234s=\frac{123}{4}
12b9n+45=−219n+45=-21n=−223n=-\frac{22}{3}

Worked examples

Problem 3a

−185=−3r−4(−5r+8)=−3r+20r−32−185=17r−32−153=17rr=−9\begin{aligned} -185&=-3r-4(-5r+8)\\ &=-3r+20r-32\\ -185&=17r-32\\ -153&=17r\\ r&=-9 \end{aligned}

Check: −3(−9)−4[−5(−9)+8]=27−212=−185-3(-9)-4[-5(-9)+8]=27-212=-185.

Problem 9b

−4f+12(4f−5)=−19−4f+2f−52=−19−2f=−332f=334\begin{aligned} -4f+\frac12(4f-5)&=-19\\ -4f+2f-\frac52&=-19\\ -2f&=-\frac{33}{2}\\ f&=\frac{33}{4} \end{aligned}

Problem 11a

−6p−8(4p+8)=98−38p−64=98−38p=162p=−8119\begin{aligned} -6p-8(4p+8)&=98\\ -38p-64&=98\\ -38p&=162\\ p&=-\frac{81}{19} \end{aligned}

Final check

All 24 values above satisfy their original equations. Fractions are left exact rather than rounded, because substitution then verifies both sides without approximation error.

Evidence boundary

The equations are reconstructed from the indexed Lesson 3.4 exercise and independently solved by substitution. The linked textbook sources support the solving method, not the wording or numbering of this worksheet.

Sources

These references support the concepts and methods used in the explanation above.

Lesson 3.4 Solving Complex 1-Variable Equations Answer Key | Verla