# 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 $x=-\frac{9}{2}$ .

Mr Hernaez runs his car at a regular speed of 50 kph and Mr Ranola at 36 kph. They started at the same place at 5:30 am and took opposite directions. At what time were they 129 km apart?
Melody wants to sell bags of mixed candy at her lemonade stand. She will mix chocolate pieces that cost $4.89 per bag with peanut butter pieces that cost$3.79 per bag to get a total of twenty-five bags of mixed candy. Melody wants the bags of mixed candy to cost her $4.23 a bag to make. How many bags of chocolate pieces and how many bags of peanut butter pieces should she use? Jake Reply enrique borrowed$23,500 to buy a car he pays his uncle 2% interest on the $4,500 he borrowed from him and he pays the bank 11.5% interest on the rest. what average interest rate does he pay on the total$23,500
13.5
Pervaiz
Amber wants to put tiles on the backsplash of her kitchen counters. She will need 36 square feet of tiles. She will use basic tiles that cost $8 per square foot and decorator tiles that cost$20 per square foot. How many square feet of each tile should she use so that the overall cost of the backsplash will be $10 per square foot? Bridget Reply The equation P=28+2.54w models the relation between the amount of Randy’s monthly water bill payment, P, in dollars, and the number of units of water, w, used. Find the payment for a month when Randy used 15 units of water. Bridget help me understand graphs Marlene Reply what kind of graphs? bruce function f(x) to find each value Marlene I am in algebra 1. Can anyone give me any ideas to help me learn this stuff. Teacher and tutor not helping much. Marlene Given f(x)=2x+2, find f(2) so you replace the x with the 2, f(2)=2(2)+2, which is f(2)=6 Melissa if they say find f(5) then the answer would be f(5)=12 Melissa I need you to help me Melissa. Wish I can show you my homework Marlene How is f(1) =0 I am really confused Marlene what's the formula given? f(x)=? Melissa It shows a graph that I wish I could send photo of to you on here Marlene Which problem specifically? Melissa which problem? Melissa I don't know any to be honest. But whatever you can help me with for I can practice will help Marlene I got it. sorry, was out and about. I'll look at it now. Melissa Thank you. I appreciate it because my teacher assumes I know this. My teacher before him never went over this and several other things. Marlene I just responded. Melissa Thank you Marlene -65r to the 4th power-50r cubed-15r squared+8r+23 ÷ 5r WENDY Reply State the question clearly please Rich write in this form a/b answer should be in the simplest form 5% August Reply convert to decimal 9/11 August 0.81818 Rich 5/100 = .05 but Rich is right that 9/11 = .81818 Melissa Equation in the form of a pending point y+2=1/6(×-4) Jose Reply write in simplest form 3 4/2 August definition of quadratic formula Ahmed Reply From Google: The quadratic formula, , is used in algebra to solve quadratic equations (polynomial equations of the second degree). The general form of a quadratic equation is , where x represents a variable, and a, b, and c are constants, with . A quadratic equation has two solutions, called roots. Melissa what is the answer of w-2.6=7.55 What Reply 10.15 Michael w = 10.15 You add 2.6 to both sides and then solve for w (-2.6 zeros out on the left and leaves you with w= 7.55 + 2.6) Korin Nataly is considering two job offers. The first job would pay her$83,000 per year. The second would pay her $66,500 plus 15% of her total sales. What would her total sales need to be for her salary on the second offer be higher than the first? Mckenzie Reply x >$110,000
bruce
greater than $110,000 Michael Estelle is making 30 pounds of fruit salad from strawberries and blueberries. Strawberries cost$1.80 per pound, and blueberries cost $4.50 per pound. If Estelle wants the fruit salad to cost her$2.52 per pound, how many pounds of each berry should she use?
$1.38 worth of strawberries +$1.14 worth of blueberries which= $2.52 Leitha how Zaione is it right😊 Leitha lol maybe Robinson 8 pound of blueberries and 22 pounds of strawberries Melissa 8 pounds x 4.5 = 36 22 pounds x 1.80 = 39.60 36 + 39.60 = 75.60 75.60 / 30 = average 2.52 per pound Melissa 8 pounds x 4.5 equal 36 22 pounds x 1.80 equal 39.60 36 + 39.60 equal 75.60 75.60 / 30 equal average 2.52 per pound Melissa hmmmm...... ? Robinson 8 pounds x 4.5 = 36 22 pounds x 1.80 = 39.60 36 + 39.60 = 75.60 75.60 / 30 = average 2.52 per pound Melissa The question asks how many pounds of each in order for her to have an average cost of$2.52. She needs 30 lb in all so 30 pounds times $2.52 equals$75.60. that's how much money she is spending on the fruit. That means she would need 8 pounds of blueberries and 22 lbs of strawberries to equal 75.60
Melissa
good
Robinson
👍
Leitha
thanks Melissa.
Leitha
nawal let's do another😊
Leitha
we can't use emojis...I see now
Leitha
Sorry for the multi post. My phone glitches.
Melissa
Vina has $4.70 in quarters, dimes and nickels in her purse. She has eight more dimes than quarters and six more nickels than quarters. How many of each coin does she have? Mckenzie Reply 10 quarters 16 dimes 12 nickels Leitha A private jet can fly 1,210 miles against a 25 mph headwind in the same amount of time it can fly 1,694 miles with a 25 mph tailwind. Find the speed of the jet. Crispy Reply wtf. is a tail wind or headwind? Robert 48 miles per hour with headwind and 68 miles per hour with tailwind Leitha average speed is 58 mph Leitha Into the wind (headwind), 125 mph; with wind (tailwind), 175 mph. Use time (t) = distance (d) ÷ rate (r). since t is equal both problems, then 1210/(x-25) = 1694/(×+25). solve for x gives x=150. bruce the jet will fly 9.68 hours to cover either distance bruce Riley is planning to plant a lawn in his yard. He will need 9 pounds of grass seed. He wants to mix Bermuda seed that costs$4.80 per pound with Fescue seed that costs $3.50 per pound. How much of each seed should he buy so that the overall cost will be$4.02 per pound?
33.336
Robinson