1.2 Linear Equations in One Variable

• Identity vs. Conditional Equations• Solve Linear Equations by Producing Equivalent

Equations• Solve Equations that Contain Fractions by

Multiplying Both Sides by LCD• Find Intercepts Algebraically

Identity vs. Conditional

• An IDENTITY is an equation that works for every number.

• For example, x2 – 9 = (x + 3)(x – 3) works for any value for x. Let’s try a few:

So how would you recognize an “identity” on an exam?

• If you are simplifying an equation and you end up with the exact same thing on both sides of the equal sign, it’s an identity.

• If you were simplifying the equation 4x+2x= 6x, and you combine like terms on the left-hand side. You’d have:

• So the answer would be “IDENTITY.”

If it is not an identity, it is a …

• Conditional Equation, which means the equation works for only some or no values for x.

• For example, x2 – 9 = 0 is only true for x = 3 or x = -3.

• (So a conditional equation does not work for all values for x.)

Linear Equation with only one Variable in it

• This might look like ax + b = 0.• Example: 2x – 4 = 0, Let’s solve it:

Notice, this is conditional because only 2 works for x.

Solve 5x+4=3x-8

• Subtract 3x from both sides.

• Subtract 4 from both sides.

• Divide both sides by 2.

Solve 6(x-1)+4=3(7x+1)

• Distribute first on this one.

Solve 24

3

3xx

To get rid of fractional coefficients, multiply both sides (every single term) by their lowest common denominator. Here, LCD = 12

Same strategy: 4

6

2

3

2

12

x

x

xx1. Factor all denominators.

2. To figure out what the LCD is, list all the factors that appear in denominators: ( )( ) (If any term is repeated in the same term’s denominator, then it will also need to be repeated in the LCD.)

3. Multiply EVERY TERM by the LCD.

4. Cancel vertically in each term.

Let’s do it: 4

6

2

3

2

12

x

x

xx

Cool that we got a nice answer of -2 BUT..

• Look at the original problem 4

6

2

3

2

12

x

x

xx

-2 is called an “extraneous solution.”

finding intercepts

• What does an x-intercept look like?

• What does a y-intercept look like?

• To find x-intercepts, set y equal to zero and solve.

• To find y-intercepts, set x equal to zero and solve.

Find the x- and y-intercepts of (8x/3) + 5 – 2y = 0

____(0, )____ ____( ,0)___

Here’s one from HW: Solve for x: 5+ax=12-bx

• One way to start is to add bx to both sides.

The key step to that last one was factoring out an x.

People get stuck there, forgetting what to do next. So if you end up with the x (or whatever you are solving for) appearing in more than one term, then get them on the same side, and factor it out.