Section 1
Use a General Strategy to Solve Linear Equations
Determine whether the given values are solutions to the equation.$6 y+10=12 y$(a) $y=\frac{5}{3}$(b) $y=-\frac{1}{2}$
Determine whether the given values are solutions to the equation.$4 x+9=8 x$(a) $x=-\frac{7}{8}$ (b) $x=\frac{9}{4}$
Determine whether the given values are solutions to the equation.$8 u-1=6 u$(a) $u=-\frac{1}{2}$ (b) $u=\frac{1}{2}$
Determine whether the given values are solutions to the equation.$9 v-2=3 v$(a) $v=-\frac{1}{3}$ (b) $v=\frac{1}{3}$
Solve each linear equation.$$15(y-9)=-60$$
Solve each linear equation.$$-16(3 n+4)=32$$
Solve each linear equation.$$-(w-12)=30$$
Solve each linear equation.$$-(t-19)=28$$
Solve each linear equation.$$51+5(4-q)=56$$
Solve each linear equation.$$-6+6(5-k)=15$$
Solve each linear equation.$$3(10-2 x)+54=0$$
Solve each linear equation.$$-2(11-7 x)+54=4$$
Solve each linear equation.$$\frac{2}{3}(9 c-3)=22$$
Solve each linear equation.$$\frac{3}{5}(10 x-5)=27$$
Solve each linear equation.$$\frac{1}{5}(15 c+10)=c+7$$
Solve each linear equation.$$\frac{1}{4}(20 d+12)=d+7$$
Solve each linear equation.$$3(4 n-1)-2=8 n+3$$
Solve each linear equation.$$9(2 m-3)-8=4 m+7$$
Solve each linear equation.$$12+2(5-3 y)=-9(y-1)-2$$
Solve each linear equation.$$-15+4(2-5 y)=-7(y-4)+4$$
Solve each linear equation.$$5+6(3 s-5)=-3+2(8 s-1)$$
Solve each linear equation.$$-12+8(x-5)=-4+3(5 x-2)$$
Solve each linear equation.$$4(p-4)-(p+7)=5(p-3)$$
Solve each linear equation.$$3(a-2)-(a+6)=4(a-1)$$
Solve each linear equation.$$4[5-8(4 c-3)]=12(1-13 c)-8$$
Solve each linear equation.$$5[9-2(6 d-1)]=11(4-10 d)-139$$
Solve each linear equation.$$3[-9+8(4 h-3)]=2(5-12 h)-19$$
Solve each linear equation.$$3[-14+2(15 k-6)]=8(3-5 k)-24$$
Solve each linear equation.$$5[2(m+4)+8(m-7)]=2[3(5+m)-(21-3 m)]$$
Solve each linear equation.$$10[5(n+1)+4(n-1)]=11[7(5+n)-(25-3 n)]$$
Classify each equation as a conditional equation, an identity, or a contradiction and then state the solution.$$23 z+19=3(5 z-9)+8 z+46$$
Classify each equation as a conditional equation, an identity, or a contradiction and then state the solution.$$15 y+32=2(10 y-7)-5 y+46$$
Classify each equation as a conditional equation, an identity, or a contradiction and then state the solution.$$18(5 j-1)+29=47$$
Classify each equation as a conditional equation, an identity, or a contradiction and then state the solution.$$24(3 d-4)+100=52$$
Classify each equation as a conditional equation, an identity, or a contradiction and then state the solution.$$22(3 m-4)=8(2 m+9)$$
Classify each equation as a conditional equation, an identity, or a contradiction and then state the solution.$$30(2 n-1)=5(10 n+8)$$
Classify each equation as a conditional equation, an identity, or a contradiction and then state the solution.$$7 v+42=11(3 v+8)-2(13 v-1)$$
Classify each equation as a conditional equation, an identity, or a contradiction and then state the solution.$$18 u-51=9(4 u+5)-6(3 u-10)$$
Classify each equation as a conditional equation, an identity, or a contradiction and then state the solution.$$45(3 y-2)=9(15 y-6)$$
Classify each equation as a conditional equation, an identity, or a contradiction and then state the solution.$$60(2 x-1)=15(8 x+5)$$
Classify each equation as a conditional equation, an identity, or a contradiction and then state the solution.$$9(14 d+9)+4 d=13(10 d+6)+3$$
Classify each equation as a conditional equation, an identity, or a contradiction and then state the solution.$$11(8 c+5)-8 c=2(40 c+25)+5$$
Solve each equation with fraction coefficients.$$\frac{1}{4} x-\frac{1}{2}=-\frac{3}{4}$$
Solve each equation with fraction coefficients.$$\frac{3}{4} x-\frac{1}{2}=\frac{1}{4}$$
Solve each equation with fraction coefficients.$$\frac{5}{6} y-\frac{2}{3}=-\frac{3}{2}$$
Solve each equation with fraction coefficients.$$\frac{5}{6} y-\frac{1}{3}=-\frac{7}{6}$$
Solve each equation with fraction coefficients.$$\frac{1}{2} a+\frac{3}{8}=\frac{3}{4}$$
Solve each equation with fraction coefficients.$$\frac{5}{8} b+\frac{1}{2}=-\frac{3}{4}$$
Solve each equation with fraction coefficients.$$2=\frac{1}{3} x-\frac{1}{2} x+\frac{2}{3} x$$
Solve each equation with fraction coefficients.$$2=\frac{3}{5} x-\frac{1}{3} x+\frac{2}{5} x$$
Solve each equation with fraction coefficients.$$\frac{1}{3} w+\frac{5}{4}=w-\frac{1}{4}$$
Solve each equation with fraction coefficients.$$\frac{1}{2} a-\frac{1}{4}=\frac{1}{6} a+\frac{1}{12}$$
Solve each equation with fraction coefficients.$$\frac{1}{3} b+\frac{1}{5}=\frac{2}{5} b-\frac{3}{5}$$
Solve each equation with fraction coefficients.$$\frac{1}{3} x+\frac{2}{5}=\frac{1}{5} x-\frac{2}{5}$$
Solve each equation with fraction coefficients.$$\frac{1}{4}(p-7)=\frac{1}{3}(p+5)$$
Solve each equation with fraction coefficients.$$\frac{1}{5}(q+3)=\frac{1}{2}(q-3)$$
Solve each equation with fraction coefficients.$$\frac{1}{2}(x+4)=\frac{3}{4}$$
Solve each equation with fraction coefficients.$$\frac{1}{3}(x+5)=\frac{5}{6}$$
Solve each equation with fraction coefficients.$$\frac{4 n+8}{4}=\frac{n}{3}$$
Solve each equation with fraction coefficients.$$\frac{3 p+6}{3}=\frac{p}{2}$$
Solve each equation with fraction coefficients.$$\frac{3 x+4}{2}+1=\frac{5 x+10}{8}$$
Solve each equation with fraction coefficients.$$\frac{10 y-2}{3}+3=\frac{10 y+1}{9}$$
Solve each equation with fraction coefficients.$$\frac{7 u-1}{4}-1=\frac{4 u+8}{5}$$
Solve each equation with fraction coefficients.$$\frac{3 v-6}{2}+5=\frac{11 v-4}{5}$$
Solve each equation with decimal coefficients.$$0.4 x+0.6=0.5 x-1.2$$
Solve each equation with decimal coefficients.$$0.7 x+0.4=0.6 x+2.4$$
Solve each equation with decimal coefficients.$$0.9 x-1.25=0.75 x+1.75$$
$$0.9 x-1.25=0.75 x+1.75$$$$1.2 x-0.91=0.8 x+2.29$$
Solve each equation with decimal coefficients.$$0.05 n+0.10(n+8)=2.15$$
Solve each equation with decimal coefficients.$$0.05 n+0.10(n+7)=3.55$$
Solve each equation with decimal coefficients.$$0.10 d+0.25(d+5)=4.05$$
Solve each equation with decimal coefficients.$$0.10 d+0.25(d+7)=5.25$$
Micah has 74 feet of fencing to make a dog run in his yard. He wants the length to be 2.5 feet more than the width. Find the length, $L,$ by solving the equation $2 L+2(L-2.5)=74$.
Paula bought $\$ 22.82$ worth of 49 -cent stamps and 21 -cent stamps. The number of 21 -cent stamps was eight less than the number of 49-cent stamps. Solve the equation $0.49 s+0.21(s-8)=22.82$ for $s,$ to find the number of 49 -cent stamps Paula bought.
Using your own words, list the steps in the general strategy for solving linear equations.
Explain why you should simplify both sides of an equation as much as possible before collecting the variable terms to one side and the constant terms to the other side.
What is the first step you take when solving the equation $3-7(y-4)=38 ?$ Why is this your first step?
If an equation has several fractions, how does multiplying both sides by the LCD make it easier to solve?
If an equation has fractions only on one side, why do you have to multiply both sides of the equation by the LCD?
For the equation $0.35 x+2.1=3.85$, how do you clear the decimal?