Problem 35 - Understanding Elementary Algebra with Geometry

Question

In Exercises \(35-46,\) determine which, if any, of the ordered pairs listed satisfy the given equation. $$2 y-4 x=10 ; \quad(7,1),(1,7),(5,0)$$

Step-by-Step Solution

Answer

(1,7) satisfies the equation.

Step 1 - Understand the equation and the ordered pairs

The given equation is \(2y - 4x = 10\). The ordered pairs to test are \((7,1), (1,7), (5,0)\).

Step 2 - Test the ordered pair (7,1)

Substitute \(x = 7\) and \(y = 1\) into the equation:\[2(1) - 4(7) = 2 - 28 = -26\].Since \(-26\) does not equal \(10\), the pair \((7,1)\) does not satisfy the equation.

Step 3 - Test the ordered pair (1,7)

Substitute \(x = 1\) and \(y = 7\) into the equation:\[2(7) - 4(1) = 14 - 4 = 10\].Since \(10\) equals \(10\), the pair \((1,7)\) satisfies the equation.

Step 4 - Test the ordered pair (5,0)

Substitute \(x = 5\) and \(y = 0\) into the equation:\[2(0) - 4(5) = 0 - 20 = -20\].Since \(-20\) does not equal \(10\), the pair \((5,0)\) does not satisfy the equation.

Key Concepts

linear equations

A linear equation is an equation that makes a straight line when it's graphed. To recognize a linear equation, look for equations that can be written in the form of ax + by = c. Here, a, b, and c are constants and x and y are variables. In our example, the given linear equation is 2y − 4x = 10. Notice how it involves no exponents or powers, only multiplication, subtraction, and constants. Linear equations are fundamental in algebra because they model relationships between quantities. They also have applications across mathematics and sciences.

ordered pairs

Ordered pairs are a way to represent points on a coordinate plane. They are written in the form (x, y), where x represents the horizontal position, and y represents the vertical position. In the context of our exercise, the given ordered pairs are (7, 1), (1, 7), and (5, 0). Each pair is tested to see if it satisfies the linear equation 2y − 4x = 10. By substituting x and y into the equation, we can determine if an ordered pair is a solution to a given equation. Ordered pairs help in identifying specific points and their relationships in algebra and geometry.

substitution method

The substitution method is a key algebraic technique for solving equations. It involves replacing one variable with a value or expression to simplify and solve the equation. In our example, each ordered pair is substituted into the linear equation to check if it satisfies that equation. For instance, substituting (7, 1) into 2y − 4x = 10 means replacing x with 7 and y with 1, yielding 2(1) − 4(7) = 2 − 28 = -26. Since -26 is not equal to 10, (7, 1) is not a solution. Repeat this step for all given pairs to identify which one satisfies the equation.

algebra concepts for beginners

Understanding basic algebra concepts, such as linear equations, ordered pairs, and the substitution method, is essential for beginners. This includes:

• Identifying linear equations and understanding their structure
• Recognizing and working with ordered pairs to represent points on a graph
• Applying the substitution method to test if an ordered pair satisfies an equation

Each of these concepts builds a foundation for more advanced topics in mathematics. Practice with these concepts will enhance problem-solving skills and boost confidence in algebraic operations.