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Chapter 3: Problem 8

Determine whether each equation has the given ordered pair as a solution. \(y=2 x+3 ;(0,3)\)

Short Answer

Expert verified
Yes, \((0, 3)\) is a solution.

Step by step solution

01

Identify the Given Equation and Ordered Pair

The equation provided is \(y = 2x + 3\) and the ordered pair is \((0, 3)\). The objective is to determine whether this ordered pair is a solution to the equation.
02

Substitute the Ordered Pair into the Equation

Substitute \(x = 0\) and \(y = 3\) into the equation \(y = 2x + 3\). After substitution, the equation becomes \(3 = 2(0) + 3\).
03

Simplify the Equation

Simplify the equation after substitution: \(3 = 0 + 3\). This simplifies to \(3 = 3\).
04

Determine if the Equation Holds True

Since both sides of the equation are equal (\(3 = 3\)), the ordered pair \((0, 3)\) is indeed a solution to the equation \(y = 2x + 3\).

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Key Concepts

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

substitution method
The substitution method is a straightforward technique used to solve systems of equations or to verify if an ordered pair is a solution to a given equation.
It involves substituting the values of variables into the equation to check if the equality holds true.

In this exercise, we have the equation \(y = 2x + 3\) and the ordered pair \((0, 3)\). To use the substitution method, follow these steps:
  • Identify the variables in the ordered pair. Here, \((x, y) = (0, 3)\).
  • Substitute the value of \(x\) into the equation. That is, replace \(x\) with 0 in the equation \(y = 2x + 3\).
  • Substitute the value of \(y\) into the equation. Replace \(y\) with 3 in the equation \(y = 2x + 3\).
  • Simplify the equation to see if both sides are equal.
This method helps determine if the given ordered pair satisfies the equation.
ordered pairs
Ordered pairs are a fundamental concept in coordinate geometry. An ordered pair is a set of values that show a specific location on a coordinate plane. The pair is written in the form \((x, y)\), where \(x\) is the horizontal coordinate and \(y\) is the vertical coordinate.

For example, the ordered pair \((0, 3)\) means:
  • Move 0 units along the x-axis (horizontal direction).
  • Move 3 units along the y-axis (vertical direction).
Ordered pairs are crucial in graphing linear equations and in solving problems that involve such equations as we saw in the given example where we checked if \((0, 3)\) is a solution to the equation \(y = 2x + 3\).
verifying solutions
Verifying solutions is the process of checking if a given ordered pair satisfies an equation. It ensures that the equation holds true for the provided values of variables.

To verify a solution, follow these steps:
  • Substitute the values of the variables from the ordered pair into the equation.
  • Simplify the equation to check both sides of the equality.
  • If both sides are equal, then the ordered pair is a solution to the equation.
In our exercise, we substituted \(x = 0\) and \(y = 3\) into the equation \(y = 2x + 3\) and simplified to verify that \(3 = 3\). This confirmed that \((0, 3)\) is indeed a solution.
Verifying solutions is essential to ensure the correctness of ordered pairs concerning specific equations.

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Most popular questions from this chapter

Graph the equations $$\begin{array}{ll}{y_{1}=1.4 x+2,} & {y_{2}=0.6 x+2} \\\\{y_{3}=1.4 x+5,} & {\text { and } \quad y_{4}=0.6 x+5}\end{array}$$ using a graphing calculator. If possible, use the Simultaneous mode so that you cannot tell which equation is being graphed first. Then decide which line corresponds to each equation.

Find the domain of \(f\). $$f(x)=\frac{7}{6-x}$$

For Exercises 123 and \(124,\) let \(f(x)=3 x^{2}-1\) and \(g(x)=2 x+5\). $$\text { Find } f(g(-4)) \text { and } g(f(-4))$$.

Use a graphing calculator to find the function values. $$h(n)=8-n-\frac{1}{n}$$ a) \(h(0.2)\) b) \(h\left(-\frac{1}{4}\right)\)

To prepare for Section \(3.8,\) review evaluating algebraic expressions (Section \(1.8)\) $$ 2-x, \text { for } x=-3 $$
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