9.4 Solving Equations

give examples of equations that are false

1 = 0 3 + 2 = 6

give examples of equations that are true.....

3+ 2 = 5 6x 3 = 3x 6 10=5 x2

the very last equation x =3 tells us that

all the equations, including the original one, have the solution 3

each equation has the same solution. why?

because a solution is a number that makes both sides equal and the equality doesn't change throughout the process.

to solve an equation involving variables means to

determine those values for the variables that make the equation true.

we can change the equation 3x = 4x + 2 = 1 to the equation

=x + 2 = 1 because according to the distributive property 3x=4x=(3-4)x = (-1)x = -x

look at figure 9.26 to see how these ways of thinking are used to solve the equations 4x + 2 + x = 5 + 3x + 3

the figure shows a step by step process of changing the equation to new equations each equation has the same solution

in this section focus on

the fundamental ideas we need to make sense of the methods we use to solve equations

if the two sides are equal and you add the same amount to both sides or take the same amount away from both sides,

then the two sides remain equal and vice versa

although there are general techniques for solving equations, sometimes we can solve an equation just by

thinking about what the equation means and by thinking about the relationship between the expressions on both sides of the equal sign

a pan balance doesn't work well for some equations, such as

those involving negative numbers. however the strategy we used by taking a pan balance view of equations works in general for all equations

one source of difficulty in solving equations is

understanding that the equals sign does not mean calculate the answer.

how can we change an equation into a new equation that has the same solutions

use properties of arithmetic or valid ways of operating with numbers (including fractions) to change an expression on either side of the equals sign into an equivalent expression add the same quantity to both sides of the equation or subtract the same quantity from both sides of the equation. multiply both sides of the equation by the same nonzero number, or divide both sides of the equation by the same nonzero number

how can we solve equations

we can reason about relationships to determine solutions to equations

to solve 2x = 6 to solve x + 3 = 7

we can think: 2 times what number is 6? the solution is 3, which we could also find by dividing x = 6/2 = we can think: what number plus 3 is 7? the solution is 4, which we could also find by subtracting x = 7 = 3 = 4

why do items 1 , 2, and 3 change an equation into a new equation that has the same solutions as the original one?

when we use item 1 we don't really change the equation at all, we just write the expression on one side of the equal sign in a different way to see why items 2 and 3 should produce a new equation that has the same solutions as the original one, remember that the solutions of an equation make the two sides equal

we can change the equation 5x = 3 to the equation

x = 5/3 by dividing both sides of the equation by 5. remember that 5x stands for 5 * x, so when we divide this by 5, we get x

in genera to solve an equation,

we change the original equation to new equations that have the same solution until we arrive at an equation that is very easy to solve, such as x=/23 or x = -7

in order to understand how to solve equations..

we must understand that we want to make the expressions to the left and right of the equal sign equal to each other

we can change the equation 6x-2 =3 to the equation

6x - 2 + 2 = 3 +2, which becomes 6x = 5

even young children in elementary school learns to solve simple equations. ex: first or second graders could be asked to

fill in the box to make the following equations true: 5 + (. ) = 7

most equations with variables are true for some values of the variable and false for other values of the variable ex, the equation. 3 + x = 5

is true when x =2 , but is false for all other values of.x

when children are asked to fill in the box to make the equation 5 + 3 = (. ) + 2 , true

many will fill in the number 8 because 5 + 3 = 8

to solve the equation 3x =12

means to find all those values for x for which 3x is equal to 12. only 3 x4 is equal to 12; 3 times any other number other than 4 is not equal to 12. so, there is only one solution to 3x=12, namely x=4

what Is a pan balance

one helpful piece of imagery for understanding equations

if we view the equation 5 + 3 = (. ) + 2 in terms of a pan balance, each side of the equation.... to solve the equation ....

page 385 each side of the equation corresponds to a side of the pan balance. to solve the equation means to make the pans balance rather than tilt to one side or the other

in algebra, we use these key ways of thinking to develop general equation-solving strategies we take a pan balance view of equations to easiest equations to solve are like x = 3 and x =2/5 because starting from a messy equation...

see ways to change an equation to a new equation that has the same solutions because they telll you what their solutions are starting from a messy equation, we try to change it step by step into an equation that is easy to solve

the values for the variables that make the equation true are called

solutions to the equation

a major part of algebra is

solving equations

what does it mean to solve an equation and what are the solutions of an equation?

some equations are true