Question

Use substitution to solve the system. $$\begin{aligned}&2 x-3 y=10\\\&3 x+3 y=15\end{aligned}$$

Short answer

The solution to the system of equations is \(x = 5, y = 0\)

Step-by-step solution

Isolate One Variable

Isolate variable \(x\) in the second equation. To do this, divide every term of the second equation by 3. The isolated equation for \(x\) will be \(x = 5 + y\).

Substitute Isolated Variable in the Other Equation

Substitute \(x = 5 + y\) from the second equation into the first equation, replacing all \(x\)s with \(5 + y\). So, the first equation becomes: \(2(5 + y) - 3y = 10\). Simplify to find the value of \(y\).

Solve for the other Variable

After substitution and simplification, the equation becomes \(10 + 2y - 3y = 10\), which simplifies further to \(-y = 0\). Thus, \(y = 0\).

Substitute \(y = 0\) into the x-isolated Equation

Substitute \(y = 0\) into the equation \(x = 5 + y\), which becomes \(x = 5 + 0 = 5\).

Key concepts

Algebraic Substitution

Mastering algebraic substitution is a fundamental skill in algebra, especially when tackling systems of equations. Imagine substitution as a swap meet for variables: you exchange one set of variables for another to simplify the problem. By replacing variables with equivalent expressions, algebraic substitution transforms complex equations into simpler ones that are much more manageable.

In the given exercise, for example, the isolated variable from one equation (\(x = 5 + y\)) takes the place of the variable in the other equation, turning \( 2 x-3 y=10 \) into a single-variable equation. This swap simplifies the system, making it easier to solve. It's like finding the perfect trade that allows you to unlock the puzzle—a key strength of algebraic substitution.

Isolating Variables

The power move in solving equations—'isolating variables'—is like playing 'hide and seek'. Your goal is to find the variable by itself, with no other variables to distract you. This step is crucial in simplifying equations and making them solvable.

Imagine the equation \( 3 x+3 y=15 \) as a mixed group of 'x' and 'y' friends at a party. Your job is to get 'x' alone by guiding the 'y' friends away. Divide the whole party by 3, and 'x' stands alone with its numerical companion, becoming \( x = 5 + y \) as seen in Step 1 of our solution. This technique is a cornerstone in algebra that enables you to solve not only simple equations but also complex systems by breaking them down into solvable pieces.

System of Equations

A 'system of equations' is a set of multiple equations that all hold true at the same time. Picture it as a team sport, where every player (equation) has to work together to win the game (find the solution). When you have two equations involving two variables, you've got a system that you can solve using various methods, with substitution being one of the most strategic plays.

Working with systems means understanding the relationships between equations and looking for the best approach to find where their paths cross, which represents the solution. It's all about seeing the big picture: finding that combo of 'x' and 'y' scores that satisfies both equations in the system, just as in the provided example where the solution is \( x = 5 \) and \( y = 0 \)—the ultimate winning move in the game.