Question

Determine whether each point is a solution of the equation. Equation Points \(y=\sqrt{x-5}\) (a) \((9,2)\) (b) \((21,4)\)

Short answer

Yes, both (9,2) and (21,4) are solutions for the equation \(y=\sqrt{x-5}\).

Step-by-step solution

Substituting the first point

Substitute the values of x and y from point (a) which is (9,2) into the equation. The equation thus becomes: '2=\sqrt{9-5}'. Simplify the right hand side to check if it equals to the left hand side.

Evaluating the results from step 1

The right hand side of the equation simplifies to \sqrt{4} which yields 2. Since this equals the left hand side of the equation, point (a) is a solution to the given equation.

Substituting the second point

Now substitute the values of x and y from point (b) which is (21,4) into the equation. The equation becomes: '4=\sqrt{21-5}'. Simplify the right hand side to check if it equals to the left hand side.

Evaluating the results from step 3

The right hand side of the equation simplifies to \sqrt{16} which yields 4. Since this equals the left hand side of the equation, point (b) is a solution to the given equation.

Key concepts

Understanding Square Roots

Square roots are mathematical expressions that indicate which number, when multiplied by itself, will result in the given value. For example, the square root of 16 is 4, because when 4 is multiplied by itself, it results in 16.
Square roots are often denoted with a radical symbol; for instance, the square root of a number \(x\) is written as \(\sqrt{x}\). One common task in algebra is simplifying square roots, involving breaking down the number under the square root symbol into a product of simpler numbers.

It's also important to note that every positive real number has two square roots: one positive (also known as the principal square root) and one negative. However, when dealing with square roots in algebra, we generally consider the principal (positive) square root.

• This distinction ensures consistency, especially when solving equations involving square roots.
• For example, \(\sqrt{4} = 2\) as the principal root and \(-2\) is also a square root of 4.

Understanding the fundamental idea of square roots is crucial when engaging in equations and solving for coordinate points effectively.

Exploring Equation Solving

Equation solving involves finding the value of an unknown variable that makes an equation true. In the given exercise, the equation \(y = \sqrt{x-5}\) requires that we plug in potential values of \(x\) and \(y\) to check if they satisfy the equation.
Here's how it essentially works:

• Substitute the potential values from a coordinate point into the equation.
• Perform any necessary arithmetic operations, such as subtraction or finding a square root.
• Compare both sides of the equation to see if they equal each other.

For instance, by substituting point (a), (9, 2), into the equation, you calculate \(\sqrt{9-5} = 2\). If the arithmetic is correct, and both sides of the equation match, the point is indeed a solution to the equation.

Equation solving is a fundamental part of algebra and often involves steps like isolating variables, performing operations inversely, and checking your work. This methodical approach helps ensure the equation holds true under the values substituted, validating the solution.

Using Coordinate Points

Coordinate points represent a position on a graph or in a plane and are commonly noted as \((x, y)\). In algebra, these points are crucial when tested against an equation to determine if they lie on the curve or graph defined by that equation.
In the context of the exercise:

• Each point, such as (9, 2) or (21, 4), consists of an \(x\) value and a \(y\) value.
• These values are substituted into the equation to check for validity.

Coordinate points are not merely numbers; they have spatial representation.
When graphing an equation like \(y = \sqrt{x-5}\), each valid coordinate point will lie on or be tangent to the curve of the graph depicted.

Understanding the role of coordinate points helps you visualize equations in graphical form, and aids in verifying whether certain points satisfy the equation, which is fundamental in analytic geometry and algebra.