Problem 40 - Elementary Algebra

Question

For the following problems, solve the linear equations in two variables. $$ -y=0 x+10, \text { if } x=3 $$

Step-by-Step Solution

Answer

Answer: When x=3, y=-10. The solution is represented as the ordered pair (3, -10).

Step 1: Plug in the given value of x in the equation

Since the problem states "if x=3", plug in this x value into the equation: -y = 0(3) + 10.

Step 2: Solve for y

Using the equation from Step 1: -y = 0(3) + 10, we can see that 0(3) on the right side equals 0, leaving: -y = 10. To solve for y, multiply both sides by -1: y = -10.

Step 3: Present the solution as an ordered pair

Now that we have the values for x and y, we can represent the solution as an ordered pair (x, y): (3, -10).

Key Concepts

Linear Equations in Two Variables

Linear equations in two variables are foundational elements in algebra and they show a relationship between two quantities. A standard form of such an equation is expressed as \( Ax + By = C \), where \( A \), \( B \), and \( C \), are constants, and \( x \) and \( y \) are the variables. Solving these equations means finding the values of the variables that make the equation true.

In the given exercise, \( -y = 0x + 10 \) is a linear equation with two variables: \( x \) and \( y \). It’s already simplified, with \( A = 0 \) and \( B = -1 \), showing there's no dependence of \( y \) on \( x \) because of the zero coefficient. This equation suggests that for any value of \( x \), the outcome for \( y \) remains constant. Understanding the structure of such equations allows students to anticipate the characteristics of their graphical representation – in this case, a horizontal line.

Substitution Method

The substitution method is a powerful technique for solving systems of linear equations. This method involves taking one equation and solving it for one variable, then 'substituting' this solution into another equation. It simplifies the process by reducing a two-variable problem to a one-variable problem.

Here’s the application of the substitution method to our exercise: We begin by taking the value for \( x \) given by the problem (\( x=3 \) and substitute it into the original equation. This simplifies the two-variable equation into one that only involves \( y \) since the \( x \) term vanishes with the coefficient of zero, offering a straightforward solution for \( y \). This illustrates how substitution can reduce complexity and lead to a solution directly.

Ordered Pairs

Ordered pairs are a way of representing solutions to equations in two variables and they describe points on a coordinate plane. An ordered pair is written in the form \( (x, y) \), where \( x \) is the value of the first variable and \( y \) is the value of the second variable. The ordered pair corresponds to the horizontal (x-axis) and vertical (y-axis) positions on a graph, respectively.

In the solution to the exercise, once the value of \( y \) is found using the substitution method, we pair it with the given \( x \) value to form the ordered pair \( (3, -10) \). This represents a specific point on the coordinate plane where these two lines would intersect if we were dealing with a system of equations. Identifying and understanding ordered pairs is crucial for analyzing linear relationships graphically and solving equations algebraically.