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Chapter 5: Problem 5

Complete the ordered pair for each equation. $$y=-6 x+10 ;(-3,)$$

Short Answer

Expert verified
The complete ordered pair for the given equation \(y = -6x + 10\) is \((-3, 28)\).

Step by step solution

01

Identify the known and unknown values

The ordered pair given is \((-3,)\), which means we are given the x-value as -3 and we need to find the corresponding y-value. Step 2: Substitute the known value into the equation
02

Substitute the x-value into the equation

We have the equation \(y = -6x + 10\), and we are given x = -3. Substitute x = -3 into the equation: \[y = -6(-3) + 10\] Step 3: Solve the equation for y
03

Solve the equation

Now, we need to solve the equation to find the value of y: \[y = -6(-3) + 10\] \[y = 18 + 10\] \[y = 28\] Step 4: Complete the ordered pair
04

Write the complete ordered pair

Now that we have found the value of y, we can complete the ordered pair: \((-3, 28)\) The complete ordered pair for the given equation \(y = -6x + 10\) is \((-3, 28)\).

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Key Concepts

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

Ordered Pairs
Ordered pairs are fundamental in representing both coordinates in geometry and solutions to equations in algebra. An ordered pair is a pair of numbers written in a specific order, usually as \(x, y\). In the context of linear equations, an ordered pair represents a point on the coordinate plane that satisfies the equation.
  • The first element, x, is called the abscissa (or x-coordinate), which indicates how far to move horizontally from the origin.
  • The second element, y, is the ordinate (or y-coordinate), which shows how far to move vertically.
Given an equation, when one of the values in an ordered pair is known, the other can be calculated by substituting the known value into the equation and solving for the unknown. In our exercise, the x-value is provided, allowing us to solve the equation for y to complete the pair.
Substitution Method
The substitution method is a technique used to solve equations, especially when dealing with ordered pairs. It's quite straightforward: you "substitute" a known value into an equation to solve for the unknown. Let's break it down.
  • Identify the known variable in the equation. Often, you'll have a partial ordered pair, like \((-3, )\), where x is provided.
  • Plug this known value into the equation. Substitute x = -3 into \(y = -6x + 10\): y = -6(-3) + 10\.
  • Solve the equation for y by performing arithmetic operations as needed. This will yield the missing part of the ordered pair.
This method is efficient for finding a complete ordered pair or value that satisfies one or more equations. In this way, we found that \((-3, 28) \) is a complete pair.
Algebraic Expressions
Algebraic expressions are combinations of variables, numbers, and operations. In this exercise, the expression is \(y = -6x + 10\), which is a linear expression because it forms a line when graphed on a coordinate plane. Here are a few key aspects about algebraic expressions:
  • Variables like x and y stand for unknowns that can change value.
  • Constants, such as -6 and 10 in our equation, are fixed numbers.
  • Coefficients, like -6, multiply the variable to scale its value.
Understanding algebraic expressions involves recognizing these components and how they interact within an equation. Solving them typically involves substituting values and performing operations to find unknown figures, as we did by using the substitution method to find y when x was -3.

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Most popular questions from this chapter

Write a system of equations and solve. A car and a truck leave the same location, the car headed east and the truck headed west. The truck's speed is 10 mph less than the speed of the car. After \(3 \mathrm{hr}\), the car and truck are \(330 \mathrm{mi}\) apart. Find the speed of each vehicle.

Graph each function. $$D(t)=3 t$$

Write a system of equations and solve. Carol orders six White Castle hamburgers and a small order of fries for \(\$ 3.91,\) and Momar orders eight hamburgers and two small fries for \(\$ 5.94 .\) Find the cost of a hamburger and the cost of an order of french fries at White Castle, (Source: White Castle menu)

The perimeter of a triangle is \(29 \mathrm{cm}\). The longest side is \(5 \mathrm{cm}\) longer than the shortest side, and the sum of the two smaller sides is \(5 \mathrm{cm}\) more than the longest side. Find the lengths of the sides of the triangle.

Find \(a\) so that \((4,3)\) is a solution to the system $$\begin{array}{l}a x+5 y=3 \\\2 x-3 y=-1\end{array}$$
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