Problem 39 - Understanding Elementary Algebra with Geometry

Question

In Exercises \(35-46,\) determine which, if any, of the ordered pairs listed satisfy the given equation. $$6 x-4 y-8=0 ; \quad(-2,-5),(6,7),(-10,-17)$$

Step-by-Step Solution

Answer

All three points satisfy the equation: (-2, -5), (6, 7), (-10, -17).

Step 1: Identify the Ordered Pairs

The given ordered pairs are (-2, -5)(6, 7)(-10, -17).

Step 2: Substitute Point (-2, -5) into the Equation

Replace x with -2 and y with -5 in the equation 6x - 4y - 8 = 0. Calculate: 6(-2) - 4(-5) - 8 = -12 + 20 - 8 = 0. Since the left side equals the right side, (-2, -5) satisfies the equation.

Step 3: Substitute Point (6, 7) into the Equation

Replace x with 6 and y with 7 in the equation 6x - 4y - 8 = 0. Calculate: 6(6) - 4(7) - 8 = 36 - 28 - 8 = 0.Since the left side equals the right side, (6, 7) satisfies the equation.

Step 4: Substitute Point (-10, -17) into the Equation

Replace x with -10 and y with -17 in the equation 6x - 4y - 8 = 0. Calculate: 6(-10) - 4(-17) - 8 = -60 + 68 - 8 = 0. Since the left side equals the right side, (-10, -17) satisfies the equation.

Key Concepts

Ordered Pairs

An ordered pair is a set of coordinates that tells us the position of a point in the coordinate plane. It's written in the form (x, y), where 'x' represents the horizontal position and 'y' represents the vertical position. When we verify solutions to linear equations, the ordered pairs help determine if specific points lie on the line represented by the equation. For example, given the equation \(6x - 4y - 8 = 0\), we need to substitute each ordered pair to see if the equation holds true.

• The given ordered pairs are (-2, -5), (6, 7), and (-10, -17).
• We will substitute these pairs into the given equation one by one to verify if they satisfy the equation.

Substitution Method

The substitution method involves replacing the variables in an equation with given values to check if the equation holds true. Let's use this method on the given equation \(6x - 4y - 8 = 0\).

• First Ordered Pair (-2, -5): Replace x with -2 and y with -5. We get 6(-2) - 4(-5) - 8 = -12 + 20 - 8 = 0. Since the left side equals the right side, (-2, -5) satisfies the equation.
• Second Ordered Pair (6, 7): Replace x with 6 and y with 7. We get 6(6) - 4(7) - 8 = 36 - 28 - 8 = 0. Since the left side equals the right side, (6, 7) satisfies the equation.
• Third Ordered Pair (-10, -17): Replace x with -10 and y with -17. We get 6(-10) - 4(-17) - 8 = -60 + 68 - 8 = 0. Since the left side equals the right side, (-10, -17) satisfies the equation.

The substitution method is a straightforward way to determine if an ordered pair is a solution to a given equation.

Solving Linear Equations

Linear equations are mathematical statements that express equality between two algebraic expressions and involve variables raised to the first power. The general form is \(ax + by + c = 0\), where 'a', 'b', and 'c' are constants.
When solving a linear equation:

• Simplify both sides of the equation (if needed).
• Replace the variable with the given values (substitution method).
• Simplify and solve to see if the equation holds true.

This can be particularly useful for verifying whether certain points or ordered pairs lie on the line represented by the linear equation.
By substituting each ordered pair and simplifying, you can check if the pair satisfies the given linear equation. This process of verifying solutions confirms whether a set of points fit the equation's criteria and lie on the graph of the line.