Problem 3 Determine whether the given poin... [FREE SOLUTION]

91影视

Elementary Algebra

John Redden

$Math Studyset 91影视 Explanations$ Math

2011 Edition

Chapter 3: Problem 3

Determine whether the given point is a solution. $$ -3 x+y=-6 ;(4,6) $$

Short Answer

Expert verified

The point (4, 6) is a solution to the equation.

Step by step solution

Identify the Equation and the Point

The given linear equation is $-3x + y = -6$ and the point provided is $(4, 6)$. The coordinates of this point are $x = 4$ and $y = 6$.

Substitute the Point into the Equation

Replace $x$ and $y$ in the equation with the coordinates from the point. By substituting, we have: $-3(4) + 6 = -6$.

Simplify the Equation

Perform the multiplication and addition within the equation: $-12 + 6 = -6$.

Check the Validity of the Equation

Simplify the left side to see if it equals the right side. We have $-12 + 6 = -6$, which simplifies to $-6 = -6$. This is a true statement.

Unlock Step-by-Step Solutions & Ace Your Exams!

Full Textbook Solutions
Get detailed explanations and key concepts
Unlimited Al creation
Al flashcards, explanations, exams and more...
Ads-free access
To over 500 millions flashcards
Money-back guarantee
We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Solution Validation

When working with linear equations, solution validation is an important step to ensure your findings are accurate. In this context, you are verifying if a given point, in this case

$ (4, 6) $

is indeed a solution to the equation

$ -3x + y = -6 $

To do this, substitute the coordinates into the equation and simplify. Begin by replacing $ x $ with 4 and $ y $ with 6:

$ -3(4) + 6 = -6 $

After simplification, ensure both sides of the equation are equal. A match indicates the point lies on the line defined by the equation. In our example:

$ -12 + 6 = -6 $
$ -6 = -6 $

Here, the equation holds true, confirming that $ (4, 6) $is a valid solution.

Substitution Method

The substitution method is a helpful technique in verifying whether a particular point solves a linear equation. This process involves the following steps for the equation

$ -3x + y = -6 $

Begin with identifying the point:

$ x = 4 $, $ y = 6 $

Then replace the variable values in the equation with the specific coordinates:

$ -3(4) + 6 = -6 $

Perform arithmetic operations involved:

First, multiply: $ -3 \times 4 = -12 $
Then, add: $ -12 + 6 = -6 $

Through substitution, you systematically determine if both sides of the equation equal each other. This confirms whether or not the point is a solution. Here, the result $ -6 = -6 $demonstrates that substitution yields a true statement, meaning the point $ (4, 6) $is indeed a solution.

Point on a Line

Understanding whether a point lies on a line defined by a linear equation is part of many mathematical tasks. A point

$ (x, y) $

rests on a line if, when substituted, the coordinates satisfy the line's equation seamlessly.
For the equation

$ -3x + y = -6 $

the point $ (4, 6) $needs to be checked. Substituting in gives:

$ -3(4) + 6 = -6 $

After calculation, the equality must hold true.
If the resulting expression after substitution confirms the equation holds, it infers the point lies precisely on the line. Here:

$ -12 + 6 = -6 $
$ -6 = -6 $

This equation confirms the point lies on the line depicted by the equation, showing that every solution accurately represents a point that sits on the line.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

91影视

Determine whether the given point is a solution. $$ -3 x+y=-6 ;(4,6) $$

Short Answer

Step by step solution

Identify the Equation and the Point

Substitute the Point into the Equation

Simplify the Equation

Check the Validity of the Equation

Key Concepts

Solution Validation

Substitution Method

Point on a Line

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Applied Mathematics

Pure Maths

Statistics

Mechanics Maths

Calculus

Study anywhere. Anytime. Across all devices.