91Ó°ÊÓ

Linear equations are algebraic expressions that illustrate a straight line when graphed. The general form of a linear equation in two variables is \( ax + by = c \). Here, \( a \), \( b \), and \( c \) are constants, while \( x \) and \( y \) are variables. These equations represent the relationship between two quantities. When you substitute specific values for \( x \) and \( y \), you can determine if the values make the equation true. This is what is called an "ordered pair," commonly represented as \((x, y)\). This exercise asks if the ordered pair \((2, -5)\) satisfies the equation \(7x + 2y = 4\). By solving the equation using these values, you can check if they form a valid solution for the equation in question.

The substitution method is a technique for solving mathematical equations, especially useful for linear equations. This method involves substituting the values of the variables into the equation to see if the expression holds true. In this exercise, you substitute \( x = 2 \) and \( y = -5 \) directly into the equation \(7x + 2y = 4\).

• Substitute \( x \) with 2, resulting in \( 7(2) \).
• Substitute \( y \) with -5, making it \( 2(-5) \).

By simplifying these expressions, you obtain \( 14 - 10 \). After calculation, both sides of the equation equal 4, confirming the ordered pair is indeed a solution. This straightforward method helps confirm or refute whether particular values solve the equation.

Algebraic solutions involve calculating and manipulating expressions to determine the validity of specific values. When you solve equations algebraically, you're performing operations like addition or multiplication to isolate variables or evaluate expressions. In the context of this problem, algebraic solutions help confirm that the equation \(7x + 2y = 4\) holds for the ordered pair \((2, -5)\). After substituting and simplifying, the equation becomes \(4 = 4\), meaning both sides are equal, thus validating the solution.

• The left-hand side of the equation represents the substituted values, \(14 - 10\).
• The right-hand side remains as the constant value, 4.

Since both sides of the equation are equal, (2, -5) is proved to be a solution. This exercise underlines the significance of solving equations algebraically to verify solutions.