Question

Tell whether the point is a solution of the equation \(4 x-y=1\). (-1,3)

Short answer

No, the point (-1, 3) is not a solution to the equation \(4x - y = 1\).

Step-by-step solution

Substitute the X and Y values

Substitute the given points into the equation. This gives us \(4*(-1) - 3\). '

Simplify the expression

Proceed by performing the operations in the equation. This results to \(-4 - 3\) which equals -7.

Compare the result

Compare the result obtained with the given result on the right side of the equation which is 1. In this case, -7 is not equal to 1.

Key concepts

Substitution Method

The substitution method is a powerful tool for solving linear equations, especially when you want to find out if a specific pair of values is a solution. In essence, the method involves replacing the variables with given numbers and simplifying the equation to see if it balances. For example, consider the equation ewline \(4x - y = 1\). When checking if the pair (-1, 3) is a solution, we substitute -1 for x and 3 for y. The result should maintain the equality if it's a valid solution.

In our step-by-step solution, we followed this method. We replaced x with -1 and y with 3, resulting in ewline \(4*(-1) - 3\). However, the simplification gave us -7, which does not equal 1, hence, (-1, 3) is not a solution to the equation. It's important to understand that for any ordered pair to be a solution to a linear equation, the substitution must result in a true statement—that is, both sides of the equation must be equal after the substitution.

Algebraic Expressions

Algebraic expressions form the backbone of algebra and represent relationships using variables and constants. They can range from simple forms like ewline \(x+5\) or \(3y\) to more complex ones such as ewline \(4x - y = 1\), which is seen in our exercise. When we work with such expressions, our goal is to manipulate them, either to simplify, factor, expand, or solve them.

In our original problem, the algebraic expression was simplified by substituting values for x and y. We took the steps to perform the arithmetic operations neatly, which lead us to the conclusion that our given point does not satisfy the equation. It's critical for students to practice these steps repetitively to become proficient in manipulating algebraic expressions, a skill that is necessary for advancing in mathematics.

Coordinate Plane

The coordinate plane, also known as the Cartesian plane, is a two-dimensional surface where we can graph equations, points, lines, and curves. It's made up of two number lines that intersect at a right angle: the horizontal axis, called the x-axis, and the vertical axis, called the y-axis. Each point on the plane is defined by an ordered pair ewline\((x, y)\), reflecting its position relative to these axes.

When determining whether the point (-1, 3) is a solution to our given linear equation, we're essentially questioning if this point lies on the line represented by the equation ewline\(4x - y = 1\) on the coordinate plane. Since our calculations demonstrated that this point does not satisfy the equation, it means that if we plotted the line and the point, they wouldn't intersect; the point would not lie on the line. Understanding how to relate algebraic equations to their graphical representations is key to grasping the full picture of linear equations and their solutions.