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Chapter 11: Problem 10

Which is a solution to \(4 x-y=8 ?\) A. \((2,2)\) B. \((-2,0)\) C. \((3,-4)\) D. \((0,-8)\)

Short Answer

Expert verified
Option D \((0, -8)\) is a solution.

Step by step solution

01

Understand the Equation

The given equation is \(4x - y = 8\). This is a linear equation in terms of \(x\) and \(y\). We need to find a solution \((x, y)\) that satisfies this equation.
02

Test Option A

Substitute \(x = 2\) and \(y = 2\) into the equation: \(4(2) - 2 = 8\). Calculate: \(8 - 2 = 6\). This does not equal 8, so \((2, 2)\) is not a solution.
03

Test Option B

Substitute \(x = -2\) and \(y = 0\) into the equation: \(4(-2) - 0 = 8\). Calculate: \(-8 = 8\). This does not hold true, so \((-2, 0)\) is not a solution.
04

Test Option C

Substitute \(x = 3\) and \(y = -4\) into the equation: \(4(3) - (-4) = 8\). Calculate: \(12 + 4 = 16\). This does not equal 8, so \((3, -4)\) is not a solution.
05

Test Option D

Substitute \(x = 0\) and \(y = -8\) into the equation: \(4(0) - (-8) = 8\). Calculate: \(0 + 8 = 8\). This equals 8, so \((0, -8)\) is a solution.

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Key Concepts

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

Solution Checking
Understanding whether a coordinate pair is a solution to a linear equation involves a process called solution checking. This process is integral in ensuring that the given coordinates satisfy the equation. Consider the linear equation we have: \( 4x - y = 8 \). Here鈥檚 how you can check the solution:
  • Substitute the Coordinates: Start by inserting the X and Y values from the coordinate pair into the equation.
  • Perform the Calculations: Carry out the arithmetic operations to simplify the equation.
  • Verify the Equality: After calculating, check if the left-hand side equals the right-hand side of the equation. If both sides are equal, the coordinate pair is a solution.
By using these steps, you ensure that no detail is overlooked and each candidate solution is thoroughly evaluated.
Substitution Method
The substitution method is a powerful technique in algebra, particularly useful in dealing with linear equations. It鈥檚 the process of replacing variables in the equation with known values to see if the equation holds true.Here鈥檚 a simple breakdown:
  • Identify Values: When testing coordinate pairs such as \((x, y)\), identify the values to substitute into the equation.
  • Replace and Solve: Substitute the value of \(x\) into the equation and solve for \(y\) or vice versa. For our equation \(4x - y = 8\), insert the value of \(x\) and then solve for \(y\).
  • Check the Result: Ensure that the mathematical operations lead to a valid equation. If they do, the pair is a solution.
Using substitution helps to concretely verify solutions, especially in exercises where multiple possible answers must be tested.
Coordinate Pairs
Coordinate pairs represent points on a graph and are used in linear equations to verify potential solutions. Each coordinate pair is made up of two values: \(x\) (the horizontal position) and \(y\) (the vertical position).Here's why they matter:
  • Represent Solutions: Each pair might be a point that lies on the graph of the equation if it satisfies the equation.
  • Graphical Interpretation: If a coordinate pair satisfies the equation, it means that point is on the line represented by the equation.
  • Testing: Solutions are tested systematically to determine whether the coordinates make the equation true.
In our example \(4x - y = 8\), any coordinate pair subject to testing must fulfill the equation when substituted, giving an insight into its accuracy and relevance in problem-solving.

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Most popular questions from this chapter

When two lines are parallel, the system is said to be: A. inconsistent B. dependent C. undefined D. consistent

Find the \(y\) intercept for \(3 x-7 y=21\) A. \((7,0)\) B. \((-3,0)\) \(C .(0,-3)\) D. \((-7,0)\)

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The slope of a horizontal line is A. 0 B. 1 C. undefined \(D,-1\)
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