An equation is a mathematical statement that shows that two expressions are equal. It consists of two sides, known as the left-hand side (LHS) and the right-hand side (RHS), separated by an equal sign (=).

There are various types of equations, including:

- Linear Equations: These equations have variables raised to the power of 1 and can be represented as y = mx + b.
- Quadratic Equations: These equations involve variables raised to the power of 2 and can be represented as ax^2 + bx + c = 0.
- Exponential Equations: These equations involve variables in the exponent position, such as a^x = b.
- Trigonometric Equations: These equations involve trigonometric functions like sine, cosine, and tangent.

To solve an equation means to find the value of the variable that makes the equation true. The goal is to isolate the variable on one side of the equation. There are different methods to solve equations, such as:

- Using the Addition Property of Equality: Adding the same value to both sides of an equation.
- Using the Subtraction Property of Equality: Subtracting the same value from both sides of an equation.
- Using the Multiplication Property of Equality: Multiplying both sides of an equation by the same value.
- Using the Division Property of Equality: Dividing both sides of an equation by the same value.

An equation can have one solution, no solution, or infinite solutions, depending on the values of the variables. When graphed, the solution to an equation is represented by the point(s) where the graph of the equation intersects the x-axis.

Now that you have a basic understanding of equations, try solving the following practice problems:

- Solve the equation 2x + 5 = 11.
- Find the solutions to the quadratic equation x^2 - 4x + 4 = 0.
- Solve the exponential equation 3^x = 81.

Equations are fundamental to mathematics and are used in various real-life applications. Understanding how to solve equations is essential for further studies in algebra and beyond.

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Algebra (NCTM)

Represent and analyze mathematical situations and structures using algebraic symbols.

Develop an initial conceptual understanding of different uses of variables.

Use symbolic algebra to represent situations and to solve problems, especially those that involve linear relationships.

Recognize and generate equivalent forms for simple algebraic expressions and solve linear equations

Grade 7 Curriculum Focal Points (NCTM)

Number and Operations and Algebra: Developing an understanding of operations on all rational numbers and solving linear equations

Students extend understandings of addition, subtraction, multiplication, and division, together with their properties, to all rational numbers, including negative integers. By applying properties of arithmetic and considering negative numbers in everyday contexts (e.g., situations of owing money or measuring elevations above and below sea level), students explain why the rules for adding, subtracting, multiplying, and dividing with negative numbers make sense. They use the arithmetic of rational numbers as they formulate and solve linear equations in one variable and use these equations to solve problems. Students make strategic choices of procedures to solve linear equations in one variable and implement them efficiently, understanding that when they use the properties of equality to express an equation in a new way, solutions that they obtain for the new equation also solve the original equation.