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Chapter 7: Problem 2

Determine whether the ordered pair (-2,1) is a solution to each equation. $$x-2 y=0$$

Short Answer

Expert verified
The ordered pair (-2,1) is not a solution to the equation \(x - 2y = 0\)

Step by step solution

01

Identify the ordered pair's x and y coordinates

The ordered pair (-2,1) denotes that x = -2 and y = 1
02

Substitute x and y values into the equation

Replace x with -2 and y with 1 in the equation \(x - 2y = 0\) to get \(-2 - 2(1) = 0\)
03

Simplify the equation

When calculated, we get \(-2 - 2 = -4\), which does not equal to zero. Therefore, (-2,1) is not a solution to the equation \(x - 2y = 0\)

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Ordered Pairs
Ordered pairs are essential one-dimensional coordinates used frequently in mathematics, especially in graphing and solving equations. Written in the format \(x, y\), each pair consists of two elements that represent specific points on a Cartesian coordinate plane.
  • The first element, \(x\), is the horizontal value, helping you move left or right from the origin.
  • The second element, \(y\), is the vertical value, helping you move up or down from the origin.
Ordered pairs play a crucial role in determining if these points satisfy certain equations. By substituting these values into an equation, you can assess whether the equation holds true for a specific point. If it does, the ordered pair is considered a solution to the equation. This method allows for a straightforward way to verify potential solutions.
The Substitution Method Explained
The substitution method is a common and useful technique for solving equations, particularly linear ones. It involves replacing variables with specific values or expressions to simplify and solve equations.
  • This can be approached by substituting a given value directly into the equation for a specific variable, as illustrated in the original exercise.
  • It can also involve solving one equation for a variable and then substituting that expression into another equation in systems of equations. However, in the scenario provided, we focused on basic substitution with ordered pairs.
By substituting values into equations, you can quickly verify the validity of ordered pairs as solutions, as demonstrated when testing \((-2, 1)\) in \(x - 2y = 0\). This method not only aids in solving equations but also enhances your understanding of how variables interact within mathematical expressions.
Basics of Linear Equations
Linear equations are among the simplest types of equations encountered in mathematics, characterized by straight line graphs on the Cartesian coordinate plane. They often involve constants and variables that are not multiplied by each other, with no powers greater than one.
  • A standard linear equation in two variables follows the form \(Ax + By = C\), where \(A\), \(B\), and \(C\) are constants.
  • The equation \(x - 2y = 0\) is a linear equation where both variables, \(x\) and \(y\), have coefficients that lead to a solution set often depicted graphically as a line.
Understanding such equations is fundamental because they serve as the building blocks for more complex algebraic concepts. Linear equations help pave the way to understanding how to balance and solve equations and are utilized in various real-world applications, from calculating interest to analyzing data trends.

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Most popular questions from this chapter

Graph the solution set of each system of inequalities, when possible. SEE EXAMPLE 6. (OBJECTIVE 4) $$\left\\{\begin{array}{l} x+y<-1 \\ x-y>-1 \end{array}\right.$$

Graph the solution. $$\left\\{\begin{array}{l}\frac{x}{2}+\frac{y}{3} \geq 2 \\\\\frac{x}{2}-\frac{y}{2}<-1\end{array}\right.$$

Graph the solution. $$y<3 x$$

Use elimination to solve each system. $$\left\\{\begin{array}{l}3(x-2)=4 y \\\2(2 y+3)=3 x\end{array}\right.$$

Explain what we mean when we say "inconsistent system."
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