Question

Solve the system by the method of substitution. $$\left\\{\begin{array}{l}x y-2=0 \\ y=\sqrt{x-1}\end{array}\right.$$

Short answer

The solution to the equations is \(x=2\) and \(y=1\).

Step-by-step solution

Identify The Substitution

As the second equation involves an expression that defined \(y\), this can be substituted into the first equation. Thus, the second equation \(y=\sqrt{x-1}\) is the ideal candidate.

Substitution Into The First Equation

Substitute \(y=\sqrt{x-1}\) into the first equation \(x y-2=0\) to get \(x(\sqrt{x-1}) - 2 = 0\). Thus, the first equation is now expressed in terms of one variable, \(x\).

Solve for x

Solve the equation \(x(\sqrt{x-1}) - 2 = 0\) yields \(x^2 - x - 2 = 0\). This can be factored to \((x-2)(x+1) = 0\), giving solutions \(x = 2\) or \(x = -1\). However, \(x = -1\) can't be used because \(\sqrt{-1-1}\) is undefined. Thus, \(x=2\) is the valid solution.

Solve for y

Substitute \(x=2\) into the second equation \(y = \sqrt{x-1}\), to get \(y=\sqrt{2-1}\), which gives \(y=1\).

Solution for the system

The solution of the system of equations is \(x = 2\) and \(y = 1\).

Key concepts

Method of Substitution

The method of substitution is a fundamental technique used to solve systems of equations, which involves replacing one variable with an equivalent expression from another equation. It's like solving a puzzle where you use the clues from one piece to fit into another. By doing this, we can transform a system of multiple variables into a single-variable equation.

When confronted with a problem where one equation in the system is isolated in terms of one variable (for example, \(y = \text{some expression}\)), you can substitute this expression into the other equation(s). This simplifies the process because you're dealing with one less variable. Subsequently, after finding the value of one variable, you can substitute back into the initial substituted equation to find the other variable's value, thus solving the system.

Systems of Equations

A system of equations is a set of two or more equations containing two or more variables. The challenge is that you must find the values of the variables that satisfy all equations simultaneously. Picture a crossroads where different paths meet—the solutions are like the spots where the paths intersect.

There are different methods to solve systems of equations, such as graphing, elimination, and substitution. Systems can have one solution (intersecting lines), no solution (parallel lines), or infinitely many solutions (the same line). Understanding the nature of the system can save time before you start solving it!

Algebraic Expressions

Algebraic expressions are the phrases of the algebra language. They are combinations of numbers, variables, and operations but without an equality sign. Think of them as incomplete sentences waiting for you to solve.

In the context of systems of equations, algebraic expressions can be manipulated, simplified, and transformed through the laws of algebra to help find the values of unknown variables. When expressions are substituted into each other, as with the substitution method, they become powerful tools for unraveling the mysteries of complex equations.

Square Roots

Square roots are like the detectives of mathematics—they help you find the number that was originally squared to get to a particular value. The square root of a number \(x\) is written as \(\sqrt{x}\), and it's asking 'What number times itself gives me \(x\)?'.

In algebra, taking the square root is an important step when solving equations. However, one must be cautious, especially when dealing with equations that require finding the square root of a variable. Remember, since a square of a real number is always positive, you can't take the square root of a negative number within real numbers—a key point when considering the potential solutions in systems of equations.