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Chapter 11: Problem 285

In the following exercises, find the ordered pairs that are solutions to the given equation. \(5 x+y=10\) (a) \((5,1)\) (b) \((2,0)\) (c) \((4,-10)\)

Short Answer

Expert verified
The solutions are (2, 0) and (4, -10).

Step by step solution

01

Identify the given equation

The given equation is: \[ 5x + y = 10 \]
02

Check ordered pair (a) (5, 1)

Substitute \( x = 5 \) and \( y = 1 \) into the equation: \[ 5(5) + 1 = 10 \]\[ 25 + 1 = 26 \]Since 26 is not equal to 10, (5, 1) is not a solution.
03

Check ordered pair (b) (2, 0)

Substitute \( x = 2 \) and \( y = 0 \) into the equation: \[ 5(2) + 0 = 10 \]\[ 10 = 10 \]Since 10 is equal to 10, (2, 0) is a solution.
04

Check ordered pair (c) (4, -10)

Substitute \( x = 4 \) and \( y = -10 \) into the equation: \[ 5(4) + (-10) = 10 \]\[ 20 - 10 = 10 \]Since 10 is equal to 10, (4, -10) is a solution.

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Key Concepts

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

Solve Linear Equations
In mathematics, solving linear equations is a fundamental skill. A linear equation is any equation that can be written in the form \[ Ax + By = C \]. It represents a straight line on a coordinate plane. In an equation like \[ 5x + y = 10 \], we aim to find values for x and y that make the equation true.
To confirm these values, we follow a simple step. Substitute the values given in ordered pairs (x, y) into the equation and see if the equation holds.
If the equation balances, the ordered pair is a solution. If it doesn't, the pair is not a solution. Let's break this down further...
Substitution Method
The substitution method is an essential algebraic technique used to solve equations.
Here, you replace a variable with a given value to verify an equation.
This technique is handy for evaluating if an ordered pair is a solution to a given equation.
For example, let's use the ordered pair (2, 0) with our equation \[ 5x + y = 10 \]:
  • First, substitute x = 2 and y = 0 into the equation.
  • We get \[ 5(2) + 0 = 10 \].
  • That simplifies to \[ 10 = 10 \], which is true.
Therefore, the pair (2, 0) is a solution. Similarly, we apply this substitution method to other pairs to check for solutions.
Coordinate Plane
A coordinate plane is a two-dimensional surface where each point is determined by a pair of numerical coordinates.
The x-axis runs horizontally, and the y-axis runs vertically. Together, they help locate points.
Every ordered pair, such as (2, 0), corresponds to a point on this plane.
In the exercise, ordered pairs are tested to see if they lie on the line represented by the equation \[ 5x + y = 10 \].
  • If an ordered pair is a solution to this equation, it means that point lies exactly on the line when plotted.
  • Using the substitution method, we verified pairs (2, 0) and (4, -10) indeed lie on this line, confirming them as solutions.

Understanding how linear equations translate to points on a coordinate plane improves your ability to visualize and solve algebraic problems.

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