Problem 36 - Algebra 1: Concepts and Skills

Question

The ordered pair \((3,5)\) is a solution of \(?\) F. \(y=5\) G. \(x=5\) H. \(y=-3\) J. \(x=-5\)

Step-by-Step Solution

Answer

The ordered pair (3,5) is a solution to the equation \(y = 5\).

Step 1: Identify the Values of the Ordered Pair

The ordered pair (3,5) is given, where 3 represents the \(x\)-value and 5 represents the \(y\)-value.

Step 2: Check the Possible Equations

Look through the possible equations to find a match for the pair. \n\n- For the equation \(y = 5\), the \(y\) value matches so this equation could be true. \n - For the equation \(x = 5\), the \(x\) value is not 5, so this equation is false.\n - For the equation \(y = -3\), the \(y\) value is not -3, so this equation is false. \n- For the equation \(x = -5\), the \(x\) value is not -5, so this equation is false.

Step 3: Determine the Correct Equation

After evaluating all possible equations, the only equation left that matches the given ordered pair (3,5) is \(y = 5\)

Key Concepts

Coordinate System

A coordinate system is like a map for mathematics. It allows us to pinpoint the location of a point by giving us two numbers that tell us how far along and up or down to go. In most cases, we use the Cartesian coordinate system, which consists of a horizontal line called the x-axis and a vertical line called the y-axis. These lines intersect at a point known as the origin, which has the coordinates (0,0).

Any point on this grid can be described using an ordered pair, like (3,5), which tells us to move 3 units along the x-axis (to the right from the origin if the number is positive) and then 5 units up the y-axis. These movements land us at the specific point we are describing on the grid. Each value in the pair has a very specific role, with the first number corresponding to the x-axis (the x-value) and the second number to the y-axis (the y-value).

Solving Equations

Solving equations is all about finding the values that make the equation true—a bit like a detective solving a mystery. When we have an equation, think of it as a balancing act, where both sides need to match up perfectly. To be good at solving equations, it's essential to understand operations like addition, subtraction, multiplication, and division, as well as knowing the order to use them in—a concept known as the order of operations.

In the context of a coordinate system, when we are given an equation like

(1) \( y = 5 \)

we are effectively being told that no matter what x-value we have, the y-value must always be 5. This is a horizontal line that crosses the y-axis at 5. If our x-value doesn't affect the outcome, then any ordered pair with a y-value of 5 will make this equation true.

X and Y Values

The x and y values in an ordered pair are like the GPS coordinates for reaching a specific location in math. The x-value, or the first number in the pair, tells us the point's position relative to the y-axis, while the y-value, the second number, indicates the position relative to the x-axis.

It's very important to remember that in the pair (3,5), 3 is the x-value and 5 is the y-value. They can't be reversed or mixed up! The x-value can be thought of as the 'across' component and the y-value the 'up or down' component in the journey to our point's location on the graph. For example, in the equation

(2) \( x = -5 \)

the x-value is set to -5, which would mean any point that lies on the vertical line passing through -5 on the x-axis has a chance of satisfying the equation. However, our ordered pair (3,5) doesn't match as the x-value here is 3, not -5.