Advance Algebra with Trigonometry, Section 2-1: Linear Equations in One Variable

Advance Algebra with Trigonometry

​Section 2-1: Linear Equations in One Variable

by Jeremy Adelmann

​Section 2-1 Objectives

​​Students will be able to:

• ​Decide whether a number is a solution of a linear equation.

• ​Solve linear equations by using the addition and multiplication properties of equality.

• ​Solve linear equations by using the distributive property.

• ​Solve linear equations with fractions or decimals.

• ​Identify conditional equations, contradictions, and indentities.

​Objective 1: Decide whether a number is a solution of a linear equations.

​If the variable in an equation can be replaced with a real number that makes the statment true, the that number is a solution of the equation.

​

​7 is a solution for the equation 

​

​An equation is solved by finding its solution set. The solution set for this equation is {7}.

​

​If multiple equations have the same solution set, then they are Equivalent Equations. The solution set is {2}

​Objective 2: Solve linear adding and mutiplying

​Solve 5x - 3x - 6 = 14 + 8x + 4.

​The goal is to isolate the variable on one side of the equation.

​

​

​Combine Like Terms

Add 6 to both sides

​

Subtract 8x from both sides

​

​Divide both sides by -6

Fill in the Blank

Solve the equation.

$$5x+2\ =\ 3x\ -\ 6$$  

Fill in the Blank

Solve the equation.

$$9p+1=7p-9$$  

Fill in the Blank

Solve the equation.

$$7x\ -5x\ +15=x+8$$  

Fill in the Blank

Solve the equation.

$$2x\ +\ 4\ -x=4x\ -\ 5$$  

​Solving a Linear Equation in One Variable

• ​Step 1: Clear Fractions - Eliminate any fractions by muliplying each side by the least common denominator.

• ​Step 2: Simplify each side separately - Use distributive property to clear parantheses and combine like terms as needed.

• ​Step 3: Isolate the variable terms on one side. Use the addition property to get all terms with variables on one side of the equation and all numbers on the other.

• ​Step 4: Isolate the varaible - Use the multiplication property to get an equation with just the variable (coefficient of 1) on one side.

• ​Step 5: Check - Substitute the proposed solution into the original equation.

​Objective 3: Using the Distibutive Property

​Solve 2(k - 5) + 3k = k + 6

• ​Since there are no fractions, Step 1 does not apply.

​

Distributive Property

Multiply

Combine Like Terms

Add 10 to both sides

​

Subtract k from both sides

​

Divide both sides by 4

Fill in the Blank

Solve each equation.

$$3\left(2t-4\right)=20-2t$$  

Fill in the Blank

Solve the equation.

$$2\left(3-2x\right)=x-4$$  

Fill in the Blank

Solve the equation.

$$2\left(x+4\right)=-4\left(x+1\right)$$  

​Obj. 4: Solving Linear Equations w/ Fractions

​Solve

​

​Start by eliminating the fractions. Multiply both sides by the LCD, 6.

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​The LCD is 6.

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​ Distributive Property

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​ Multiply;

​Obj. 4: Solving Linear Equations w/ Fractions

​

​

​

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​Distributive Property

​Multiply

​Combine Like Terms

​Add 17 to both sides

​Combine Like Terms

​Divide both sides by 7

Fill in the Blank

Solve the equation.

$$\frac{3x-1}{4}+\frac{x+3}{6}=3$$  

Fill in the Blank

Solve the equation.

$$\frac{3x+2}{7}-\frac{x+4}{5}=2$$