8.3 Solve equations with variables and constants on both sides  (Page 4/5)

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Solve: $5\left(x+3\right)=35.$

x = 4

Solve: $6\left(y-4\right)=-18.$

y = 1

Solve: $-\left(x+5\right)=7.$

Solution Simplify each side of the equation as much as possible by distributing. The only $x$ term is on the left side, so all variable terms are on the left side of the equation. Add 5 to both sides to get all constant terms on the right side of the equation. Simplify. Make the coefficient of the variable term equal to 1 by multiplying both sides by -1. Simplify. Check: Let $x=-12$ .     Solve: $-\left(y+8\right)=-2.$

y = −6

Solve: $-\left(z+4\right)=-12.$

z = 8

Solve: $4\left(x-2\right)+5=-3.$

Solution Simplify each side of the equation as much as possible. Distribute. Combine like terms The only $x$ is on the left side, so all variable terms are on one side of the equation. Add 3 to both sides to get all constant terms on the other side of the equation. Simplify. Make the coefficient of the variable term equal to 1 by dividing both sides by 4. Simplify. Check: Let $x=0$ . Solve: $2\left(a-4\right)+3=-1.$

a = 2

Solve: $7\left(n-3\right)-8=-15.$

n = 2

Solve: $8-2\left(3y+5\right)=0.$

Solution

Be careful when distributing the negative. Simplify—use the Distributive Property. Combine like terms. Add 2 to both sides to collect constants on the right. Simplify. Divide both sides by −6. Simplify. Check: Let $y=-\frac{1}{3}$ . Solve: $12-3\left(4j+3\right)=-17.$

$j=\frac{5}{3}$

Solve: $-6-8\left(k-2\right)=-10.$

$k=\frac{5}{2}$

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

Solution Distribute. Combine like terms. Subtract $3x$ to get all the variables on the right since $8>3$ . Simplify. Subtract 9 to get the constants on the left. Simplify. Divide by 5. Simplify. Check: Substitute: $-4=x$ . Solve: $6\left(p-3\right)-7=5\left(4p+3\right)-12.$

p = −2

Solve: $8\left(q+1\right)-5=3\left(2q-4\right)-1.$

q = −8

Solve: $\frac{1}{2}\left(6x-2\right)=5-x.$

Solution Distribute. Add $x$ to get all the variables on the left. Simplify. Add 1 to get constants on the right. Simplify. Divide by 4. Simplify. Check: Let $x=\frac{3}{2}$ . Solve: $\frac{1}{3}\left(6u+3\right)=7-u.$

u = 2

Solve: $\frac{2}{3}\left(9x-12\right)=8+2x.$

x = 4

In many applications, we will have to solve equations with decimals. The same general strategy will work for these equations.

Solve: $0.24\left(100x+5\right)=0.4\left(30x+15\right).$

Solution Distribute. Subtract $12x$ to get all the $x$ s to the left. Simplify. Subtract 1.2 to get the constants to the right. Simplify. Divide. Simplify. Check: Let $x=0.4$ . Solve: $0.55\left(100n+8\right)=0.6\left(85n+14\right).$

1

Solve: $0.15\left(40m-120\right)=0.5\left(60m+12\right).$

−1

Key concepts

• Solve an equation with variables and constants on both sides
1. Choose one side to be the variable side and then the other will be the constant side.
2. Collect the variable terms to the variable side, using the Addition or Subtraction Property of Equality.
3. Collect the constants to the other side, using the Addition or Subtraction Property of Equality.
4. Make the coefficient of the variable 1, using the Multiplication or Division Property of Equality.
5. Check the solution by substituting into the original equation.
• General strategy for solving linear equations
1. Simplify each side of the equation as much as possible. Use the Distributive Property to remove any parentheses. Combine like terms.
2. Collect all the variable terms to one side of the equation. Use the Addition or Subtraction Property of Equality.
3. Collect all the constant terms to the other side of the equation. Use the Addition or Subtraction Property of Equality.
4. Make the coefficient of the variable term to equal to 1. Use the Multiplication or Division Property of Equality. State the solution to the equation.
5. Check the solution. Substitute the solution into the original equation to make sure the result is a true statement.

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