91Ó°ÊÓ

In algebra, a system of equations is a set of two or more equations with the same set of unknowns. The goal is to find the values of the unknowns that satisfy each equation in the system simultaneously.

A common way to solve these systems is the substitution method, where one equation is solved for one variable in terms of others, and this expression is substituted into the other equations. This helps in breaking down complex problems step by step.

In the problem given, we have two equations:

• Equation 1: \( x+y=1 \)
• Equation 2: \( (x-1)^2 + (y+2)^2 = 10 \)

These equations form a system of equations, and by solving them using the substitution method, we can find the values of \( x \) and \( y \) that work for both.

Solving equations is all about finding the unknown values that make an equation true. When dealing with systems, we often solve them using a strategic approach like the substitution method.

The first step involves isolating a variable in one of the equations, which simplifies the problem. In our example, we isolate \( y \) from the first equation, yielding \( y = 1 - x \). This creates an expression for \( y \) based on \( x \), which can then be substituted into the second equation.

Once substituted, the original equation is simplified, allowing us to solve for the remaining unknown. This process requires careful algebraic manipulation, ensuring that each term is handled correctly to avoid errors.

Algebraic expressions are combinations of variables, numbers, and operators (such as + and -) that represent mathematical relationships. In the context of solving systems of equations, these expressions play a crucial role.

For instance, when you rewrite one of the equations in terms of one variable:

• From \( x+y=1 \), rewriting gives \( y = 1 - x \).

This creates an algebraic expression representing \( y \) with \( x \) as its component.

These expressions are then used to substitute values and solve the system. Understanding algebraic expressions is foundational because they help in forming equations and executing solutions logically.

Verification of solutions is the crucial final step when solving systems of equations. It ensures that the values obtained truly satisfy all equations in the system.

To verify, substitute the solutions back into the original equations to check if the left-hand side matches the right.

In our situation, substituting \( x = 3 \) and \( y = \frac{1}{3} \) back into both equations confirms their correctness:

• \( x + y = 1 \) becomes \( 3 + \frac{1}{3} = 1 \), which checks out when simplified.
• \( (x-1)^2 + (y+2)^2 = 10 \) becomes \( (3-1)^2 + (\frac{1}{3}+2)^2 = 10 \), correctly simplifies to \( 10 \).

This dual confirmation reassures us that the computations are accurate and the solutions are valid.