Problem 28 - Algebra 1

Question

Find three different ordered pairs that are solutions of the equation. \(y=3(6 x-1)\)

Step-by-Step Solution

Answer

The ordered pairs (1,15), (2,33), and (3,51) are all solutions to the given equation.

Step 1: Choose values for x

Firstly, select three arbitrary different values for x. Let's choose values like 1, 2, and 3.

Step 2: Compute corresponding y-values

Calculate the corresponding y-values using the equation \( y=3(6 x-1) \). For \( x = 1 \), it will be \( y = 3[6(1) - 1] = 3[6 - 1] = 3[5] = 15 \). For \( x = 2 \), calculate \( y = 3[6(2) - 1] = 3[12 - 1] = 3[11] = 33 \). And for \( x = 3 \), do the calculation \( y = 3[6(3) - 1] = 3[18 - 1] = 3[17] = 51 \).

Step 3: Construct ordered pairs

Get the ordered pairs by combining each x-value with its corresponding y-value obtained in the previous step. The pairs will be: (1, 15), (2, 33), and (3, 51).

Key Concepts

Linear Equations

Understanding linear equations is fundamental to mastering algebra. A linear equation is a type of equation that forms a straight line when graphed on a coordinate plane. It usually comes in the form of \( ax + by = c \) where \( a \) and \( b \) are coefficients and \( c \) is a constant. The equation provided in the exercise, \( y=3(6x-1) \), is also a linear equation. Despite the multiplication and subtraction inside the parentheses, this equation represents a line because it can be simplified to the standard linear form. The key characteristic of any linear equation is its consistence in the rate of change; for each unit increase or decrease in \( x \) the value of \( y \) will consistently change by a fixed amount, determined by the equation's slope.

Solving Algebraic Equations

To solve an algebraic equation means to find all its possible solutions, which are the values of the variables that make the equation true. In the exercise, the goal is to find distinct ordered pairs that satisfy the equation \( y=3(6x-1)\). The process involves substituting chosen values for \( x \) into the equation to get corresponding \( y \) values — this effectively 'solves' the equation for those specific instances. For example, when \( x = 1 \), substituting into the equation gives us \( y = 15 \), making (1, 15) a solution to the equation. When solving algebraic equations, it's important to perform each arithmetic step carefully to avoid common mistakes, such as incorrect sign changes or errors in basic operations like multiplication and addition.

Graphing Linear Equations

Visualizing algebraic expressions can greatly enhance understanding, and graphing linear equations is a perfect example. The graph of a linear equation is a straight line, and to plot it, you usually need just two points through which the line passes. But more points can provide an accurate visualization and confirm the line's slope and position. In the given exercise, after finding the ordered pairs such as (1, 15), (2, 33), and (3, 51), each pair represents a point on the graph. Placing these points on a coordinate plane and then drawing a line through them will reveal the visual representation of the equation \( y=3(6x-1) \). This linear graph helps students understand how changes in the value of \( x \) affect the value of \( y \), illustrating the concept of slope, which in this case, is very steep due to the large coefficient of \( x \) in the equation.