Linear Equations Summary

Linear Equations in One Variable

  • Learning Targets:

    • Solve linear equations in one variable.

    • Apply linear equations to various problem types (geometry, money, etc.).

  • Solving Linear Equations with Decimals:

    1. Multiply both sides by a power of 10 based on decimals involved.

    2. Solve the resulting linear equation.

  • Applications of Linear Equations:

    • Translate word problems into equations, e.g.:

    • Money problems: Identify variables and set up equations based on given conditions.

    • Mixture problems: Use proportions to find unknown quantities in mixtures.

    • Number problems: Express relationships between numbers as equations.

    • Motion problems: Relate speed, time, and distance in equations.

    • Age problems: Set up equations to find relationships between ages over time.

  • Solving Linear Equations with Fractions:

    1. Multiply both sides by the least common denominator (LCD).

    2. Solve the resulting equation.

  • Examples provided include solving equations such as:

    • 3rac43x−2rac34=rac563 rac{4}{3} x - 2 rac{3}{4} = rac{5}{6}

    • Applications related to tickets, distances apart, and age relationships.


Linear Equations in One Variable

In this section, we will cover linear equations in one variable, focusing on problem-solving and applications. The primary learning targets include the ability to solve linear equations and apply them to various problems, including geometry and financial situations.

To solve linear equations with decimals, you should multiply both sides by a power of 10 according to the decimals involved, and then solve the resulting linear equation. In contrast, for solving linear equations with fractions, multiply both sides by the least common denominator (LCD) and proceed to solve the equation.

Applications of linear equations are broad and include translating word problems into equations. For instance, money problems require identifying variables and setting up equations based on given conditions. Mixture problems involve using proportions to determine unknown quantities, while number problems express relationships between numbers as equations. Additionally, motion problems relate speed, time, and distance through equations, and age problems set up equations to find relationships between ages over time.

Examples of solving equations include instances such as 343x−234=563\frac{4}{3}x - 2\frac{3}{4} = \frac{5}{6}, along with applications related to tickets, distances apart, and age relationships.