2.4 Use a general strategy to solve linear equations

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By the end of this section, you will be able to:
• Solve equations using a general strategy
• Classify equations

Before you get started, take this readiness quiz.

1. Simplify: $\text{−}\left(a-4\right).$
If you missed this problem, review [link] .
2. Multiply: $\frac{3}{2}\left(12x+20\right)$ .
If you missed this problem, review [link] .
3. Simplify: $5-2\left(n+1\right)$ .
If you missed this problem, review [link] .
4. Multiply: $3\left(7y+9\right)$ .
If you missed this problem, review [link] .
5. Multiply: $\left(2.5\right)\left(6.4\right)$ .
If you missed this problem, review [link] .

Solve equations using the general strategy

Until now we have dealt with solving one specific form of a linear equation. It is time now to lay out one overall strategy that can be used to solve any linear equation. Some equations we solve will not require all these steps to solve, but many will.

Beginning by simplifying each side of the equation makes the remaining steps easier.

How to solve linear equations using the general strategy

Solve: $-6\left(x+3\right)=24.$

Solution

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

$x=4$

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

$y=1$

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 on one side of the equation.
Use the Addition or Subtraction Property of Equality.
3. Collect all the constant terms on 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.

Solve: $\text{−}\left(y+9\right)=8.$

Solution

 Simplify each side of the equation as much as possible by distributing. The only $y$ term is on the left side, so all variable terms are on the left side of the equation. Add $9$ to both sides to get all constant terms on the right side of the equation. Simplify. Rewrite $-y$ as $-1y$ . Make the coefficient of the variable term to equal to $1$ by dividing both sides by $-1$ . Simplify. Check: Let $y=-17$ .

Solve: $\text{−}\left(y+8\right)=-2.$

$y=-6$

Solve: $\text{−}\left(z+4\right)=-12.$

$z=8$

Solve: $5\left(a-3\right)+5=-10$ .

Solution

 Simplify each side of the equation as much as possible. Distribute. Combine like terms. The only $a$ term is on the left side, so all variable terms are on one side of the equation. Add $10$ to both sides to get all constant terms on the other side of the equation. Simplify. Make the coefficient of the variable term to equal to $1$ by dividing both sides by $5$ . Simplify. Check: Let $a=0$ .

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

$m=2$

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

$n=2$

Solve: $\frac{2}{3}\left(6m-3\right)=8-m$ .

Solution

 Distribute. Add $m$ to get the variables only to the left. Simplify. Add $2$ to get constants only on the right. Simplify. Divide by $5$ . Simplify. Check: Let $m=2$ .

Solve: $\frac{1}{3}\left(6u+3\right)=7-u$ .

$u=2$

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

$x=4$

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

Solution

 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: $4\left(x-1\right)-2=5\left(2x+3\right)+6$ .

Solution

 Distribute. Combine like terms. Subtract $4x$ to get the variables only on the right side since $10>4$ . Simplify. Subtract $21$ to get the constants on left. Simplify. Divide by 6. Simplify. Check: Let $x=-\frac{9}{2}$ .

the proudact of 3x^3-5×^2+3 and 2x^2+5x-4 in z7[x]/ is
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Tausif
Tausif
Can you explain further
p mulripied-5 and add to 30
Tausif
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TAKE OFF THE LAST DIGIT AND MULTIPLY IT 9. SUBTRACT IT THE DIGITS YOU HAVE LEFT. IF THE ANSWER DIVIDES BY 13(OR IS ZERO), THEN YOUR ORIGINAL NUMBER WILL ALSO DIVIDE BY 13!IS DIVISIBLE BY 13
BAINAMA
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snigdha
How did you work it out?
Esther
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Orlando
hello, I have algebra phobia. Subtracting negative numbers always seem to get me confused.
what do you need help in?
Felix
Heather
look at the numbers if they have different signs, it's like subtracting....but you keep the sign of the largest number...
Felix
for example.... -19 + 7.... different signs...subtract.... 12 keep the sign of the "largest" number 19 is bigger than 7.... 19 has the negative sign... Therefore, -12 is your answer...
Felix
—12
Thanks Felix.l also get confused with signs.
Esther
Thank you for this
Shatey
ty
Graham
think about it like you lost $19 (-19), then found$7(+7). Totally you lost just $12 (-12) Annushka I used to struggle a lot with negative numbers and math in general what I typically do is look at it in terms of money I have -$5 in my account I then take out 5 more dollars how much do I have in my account well-\$10 ... I also for a long time would draw it out on a number line to visualize it
Meg
practicing with smaller numbers to understand then working with larger numbers helps too and the song/rhyme same sign add and keep opposite signs subtract keep the sign of the bigger # then you'll be exact
Meg
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168.50=R
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46
mustee
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Joseph
john is 16. wanjiru is 11.
Felix
27-5=22 22÷2=11 11+5=16
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I don't see where the answers are.
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3
Christopher
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