Problem 78 - Algebra 1: Concepts and Skills

Question

Use the substitution method to solve the linear system. $$ \begin{aligned} &2 x+3 y=-5\\\ &x-2 y=-6 \end{aligned} $$

Step-by-Step Solution

Answer

The solution to the system of equations is \(x = 1.14\) and \(y = -2.43\)

Step 1: Express Variable

Choose either x or y from the equation to express in terms of the other variable. From the given equations \(2x + 3y = -5\) and \(x - 2y = -6\), the second equation is easier to rewrite in terms of x, so: \(x = 2y + 6\)

Step 2: Substitute Variable

Replace x in the first equation with the expression we got in step 1. This gives \(2(2y+6) + 3y = -5\), which simplifies to \(7y + 12 = -5\)

Step 3: Solve for y

Solving for y, we subtract 12 from both sides to get \(7y = -17\), then divide by 7 to get \(y = -17/7 = -2.43\) (approx.)

Step 4: Substitute y into the second equation

Now, we substitute y = -2.43 into the second equation \(x = 2(-2.43) + 6\), which gives \(x = 1.14\) (approx.)

Key Concepts

Linear System

A linear system, also known as a system of linear equations, consists of two or more linear equations with the same set of variables. In this exercise, our linear system contains two equations:

• \(2x + 3y = -5\)
• \(x - 2y = -6\)

The objective when solving a linear system is to find the values of the variables that satisfy all equations simultaneously.
This means we're looking for a specific point or pair of values for \(x\) and \(y\) that fits both equations perfectly.
Linear systems can have unique solutions, no solutions, or infinitely many solutions based on their graphical representations. For instance, two lines might intersect at a single point, be parallel and never intersect, or coincident and lie on top of each other.
In our given system, the method we use, substitution, will help us determine the exact point where these two lines intersect, which signifies the solution of the system.

Solving Equations

Solving equations involves finding the value of the variable that makes the equation true. In our exercise, we deal with solving equations within a linear system using the substitution method. Breaking it down:
First, select one of the equations and solve for one variable in terms of the other. For example, in the equation \(x - 2y = -6\), express \(x\) in terms of \(y\):

• \(x = 2y + 6\)

Next, substitute this expression into the other equation to solve for \(y\). You replace \(x\) in the first equation \(2x + 3y = -5\) with \(2y + 6\) and proceed to simplify and solve:

• \(2(2y+6) + 3y = -5\)
• \(7y + 12 = -5\)

Subtract 12 from both sides:

• \(7y = -17\)

Then, divide by 7 to find \(y = -17/7\).
This gives us a value for \(y\) which can then be plugged back into the equation \(x = 2y + 6\) to solve for \(x\). Finally, substitute the \(y\) value into either original equation to find \(x\).
This process ensures both variables are correctly determined and consistent with both equations.

Algebraic Expressions

Algebraic expressions are mathematical phrases that can include numbers, variables, and operators (like +, -, *, /). They form the building blocks of algebra, allowing us to represent real-world phenomena and solve complex problems. In solving the linear system by substitution, algebraic expressions are instrumental.
When you manipulate these expressions, like substituting one variable in terms of another, you transform the problem into something solvable. Let's break this process in our exercise:
First, rewrite the expression for \(x\) in terms of \(y\) from \(x - 2y = -6\), resulting in \(x = 2y + 6\).
This equation alone is an algebraic expression: it signals that wherever \(x\) appears, \(2y + 6\) can be used.

• The expression \(2(2y + 6) + 3y = -5\) is a composite expression involving distribution of multiplication and addition to simplify.
• Simplifying involves combining like terms until you isolate variables to solve equations.

This use of algebraic expressions is a common method in algebra that makes it easier to manipulate and solve complex equations, showing how concepts translate into manageable steps.