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Magazine Chapter 7: Problem 17
Find four solutions of equation. Write the solutions as ordered pairs. \(y=2 x+5\)
Short Answer
Expert verified
(-1, 3), (0, 5), (1, 7), (2, 9).
Step by step solution
01
Understand the Equation
The given equation is a linear equation in two variables, where the independent variable is \(x\) and the dependent variable is \(y\). This is the equation of a line with a slope of 2 and a y-intercept at 5.
02
Choose Values for x
We start by selecting different values for \(x\). Typically, choosing small integer values can make calculations easier. Let's choose the values \(x = -1, 0, 1,\) and \(2\).
03
Calculate Corresponding y Values
Substitute each chosen value of \(x\) into the equation \(y = 2x + 5\) to find the corresponding \(y\) values:- For \(x = -1\), \(y = 2(-1) + 5 = 3\).- For \(x = 0\), \(y = 2(0) + 5 = 5\).- For \(x = 1\), \(y = 2(1) + 5 = 7\).- For \(x = 2\), \(y = 2(2) + 5 = 9\).
04
Write Solutions as Ordered Pairs
Each solution of the equation corresponds to a point (or ordered pair) on the line. Based on our calculations, the solutions are:- \((-1, 3)\)- \((0, 5)\)- \((1, 7)\)- \((2, 9)\)
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Ordered Pairs
An ordered pair is a simple and effective way to represent points on a graph in coordinate geometry. An ordered pair consists of two elements usually written in parentheses with a comma separating them, like \((x, y)\). In this representation:
- The first element, \(x\), is the horizontal coordinate, also known as the abscissa.
- The second element, \(y\), is the vertical coordinate, also referred to as the ordinate.
Slope-Intercept Form
The slope-intercept form is one of the most widely used and simplest forms of a linear equation. It is expressed as \(y = mx + b\). In this formula:
- \(m\) represents the slope of the line. The slope indicates the rate at which \(y\) changes with respect to \(x\). For example, a slope of 2 means that for every unit \(x\) increases, \(y\) increases by 2 units.
- \(b\) is the y-intercept. This is the point where the line crosses the y-axis, meaning it is the \(y\) value when \(x\) is 0.
Independent and Dependent Variables
In any equation with two variables, one is typically independent, and the other is dependent. Here's a simple way to understand these concepts:
- The independent variable is the variable you have control over. In our linear equation of form \(y = 2x + 5\), \(x\) is the independent variable. You choose values for \(x\), and it can take any value you want within your context.
- The dependent variable is the variable whose value depends on the value of the independent variable. In the equation \(y = 2x + 5\), \(y\) changes based on the value you select for \(x\). It is 'dependent' because its values "depend" on \(x\).
Coordinate Geometry
Coordinate geometry, also known as analytic geometry, is a branch of mathematics that allows us to represent geometric shapes and relationships through algebra. In coordinate geometry:
- The placement of points on a plane is given by ordered pairs \((x, y)\).
- These points are positioned along axes, with the x-axis running horizontally and the y-axis vertically.
