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Chapter 4: Problem 7

Determine if each ordered pair is a solution of the given equation. $$y=5 x-6 ;(3,11)$$

Short Answer

Expert verified
The ordered pair \((3, 11)\) is not a solution of the given equation \(y=5x-6\) because after substituting the x and y values, the equation does not hold true: \(11 \neq 9\).

Step by step solution

01

Identify the x and y values in the ordered pair

From the given ordered pair $$(3, 11)$$, we can identify the x-value as 3 and the y-value as 11.
02

Substitute the x and y values into the equation

Now, we substitute the x and y values into the given equation, $$y = 5x - 6$$: $$11 = 5 (3) - 6$$
03

Simplify the equation

We can simplify the equation to see if it holds true: $$11 = 15 - 6$$
04

Check if the equation holds true

After simplifying the equation, we have: $$11 = 9$$ Since the equation does not hold true, we can conclude that the ordered pair $$(3, 11)$$ is not a solution to the given equation $$y = 5x - 6$$.

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Key Concepts

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

Ordered Pairs
When working with linear equations, you will often encounter ordered pairs. An ordered pair is a set of two numbers that give the coordinates of a point on a graph. The first number represents the x-coordinate, and the second number represents the y-coordinate. For example, in the ordered pair \((3, 11)\), 3 is the x-value, and 11 is the y-value.
It's like giving directions to a point on a map, where the x-value tells you how far to go horizontally and the y-value tells you how far to go vertically. Understanding ordered pairs is vital, as they are used to see if a point indeed lies on the graph of a given equation.
Solution of an Equation
In mathematics, a solution of an equation is any pair of values that makes the equation true. For linear equations in two variables, such as \(y = 5x - 6\), we want to determine if a particular ordered pair is a solution. This means checking whether the y-value and x-value from the ordered pair satisfy the equation once plugged in.
For instance, to check if \((3, 11)\) is a solution to \(y = 5x - 6\), you substitute these values into the equation. If both sides of the equation equal the same number, then the ordered pair is a solution.
Substitute Values
Substituting values means replacing the variables in an equation with specific numbers to check if the equation is true. This step is crucial in determining if an ordered pair is a solution.
Using the equation \(y = 5x - 6\) and the ordered pair \((3, 11)\), substitute 3 for \(x\) and 11 for \(y\). This process is like entering a password; if the right values are substituted, both sides of the equation should match.
  • Replace \(x\) with 3: \(5(3) - 6\)
  • Replace \(y\) with 11: \(11 = 15 - 6\)
After substitution, you start to simplify the equation to test if it holds true.
Simplifying Equations
Simplifying an equation involves performing arithmetic operations to check if the left side equals the right side. This is the final test to determine if the substituted values are a solution.
After substituting the values into \(y = 5x - 6\), you simplify the expression on the right-hand side:
  • Multiply: \(5 \times 3 = 15\)
  • Subtract: \(15 - 6 = 9\)
We see that \(11 = 9\) is not a true statement, indicating that \((3, 11)\) is not a solution to the equation. Simplifying reveals the equation doesn't hold, proving the ordered pair doesn't lie on the line described by the equation.

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Most popular questions from this chapter

The median hourly wage of an embalmer in Illinois in 2002 was \(\$ 17.82 .\) Seth's earnings, \(E\) (in dollars), for working \(t\) hr in a week can be defined by the function \(E(t)=17.82 t .\) (Source: www.igpa.uillinois.edu) a) How much does Seth earn if he works 30 hr? b) How much does Seth earn if he works 27 hr? c) How many hours would Seth have to work to make \(\$ 623.70 ?\) d) If Seth can work at most 40 hr per week, what is the domain of this function? e) Graph the function.

Determine the domain of each relation, and determine whether each relation describes \(y\) as a function of \(x .\) $$y=\frac{7}{6 x}$$

Graph each function. $$N(t)=3.5 t+1$$

Graph each function using the slope and \(y\) -intercept. $$g(x)=2 x+\frac{1}{2}$$

Graph each function by finding the \(x\) - and \(y\) -intercepts and one other point. $$f(x)=\frac{1}{3} x+1$$
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