Problem 50 - Algebra A Combined Function

Question

As you solve the system \(\left\\{\begin{array}{c}3 x-y=-6 \\ -3 x+2 y=7\end{array},\right.\) you ind that \(y=1\). Is this the solution to the system?

Step-by-Step Solution

Answer

Yes, \(y = 1\) is the correct solution for this system.

Step 1: Solve for x using y = 1

We have the system of equations: 1. \(3x - y = -6\)2. \(-3x + 2y = 7\)First, let's substitute \(y = 1\) into the first equation:\(3x - 1 = -6\)Add 1 to both sides:\(3x = -5\)Divide both sides by 3 to solve for \(x\):\(x = -\frac{5}{3}\).

Step 2: Verify the solution with the second equation

Now, substitute \(x = -\frac{5}{3}\) and \(y = 1\) into the second equation:\(-3 \left(-\frac{5}{3}\right) + 2(1) = 7\)Simplify:\(5 + 2 = 7\)\(7 = 7\), which is true.

Step 3: Confirm the validity of the solution

Since both equations are satisfied with \(x = -\frac{5}{3}\) and \(y = 1\), \(y = 1\) is indeed part of the solution to the system.

Key Concepts

Solving Algebraic Equations

When it comes to solving algebraic equations, you're essentially trying to find values for the variables that make the equations true. An algebraic equation is a mathematical statement where two expressions are set equal to each other. For instance, in the equation \(3x - y = -6\), \(x\) and \(y\) are the variables we need to solve for.

To solve such equations, follow these steps:

• Identify what you are solving for. In systems of equations, you'll typically have more than one variable.
• Isolate one of the variables. This often involves using mathematical operations like addition, subtraction, multiplication, or division.
• Substitute the isolated variable, if necessary, into another equation to find the other variable(s).
• Check your solution by plugging the values back into the original equation(s) to ensure they make each equation true.

Mastering algebraic equations involves patience and practice. By understanding the process and working through each step methodically, solving equations becomes manageable.

Substitution Method

The substitution method is a powerful tool used to solve systems of equations. The idea is simple: solve one of the equations for one variable, and then substitute that expression into the other equation. This transforms a two-variable system into a single-variable equation, which can often be solved easily.

Here's how you can apply the substitution method:

• Begin by solving one of the equations for one of the variables. For example, solve for \(y\) in terms of \(x\) or vice versa, depending on which makes the math simpler.
• Once you have one variable expressed in terms of the other, substitute that expression into the other equation.
• This results in an equation with just one variable. Solve this equation to find the value of that variable.
• With this value, go back to your earlier expression and find the value of the other variable.

Using the substitution method can simplify solving systems by focusing on one variable at a time. It's a helpful technique especially when equations are already simplified to an easily manageable form.

Linear Equations

Linear equations are the building blocks of algebra. They describe straight lines when graphed on a coordinate plane. A linear equation is in the form of \(ax + by = c\), where \(a\), \(b\), and \(c\) are constants. In a system of linear equations, you're dealing with multiple lines, and you're interested in finding where they intersect.

Key features of linear equations include:

• They have constant coefficients for their variables \(x\) and \(y\).
• Graphically, they represent straight lines.
• The solution to a linear system is the point(s) where the lines intersect on the graph.
• These intersections represent the values of the variables that satisfy all equations in the system.

Understanding linear equations and how they interact with one another in a system is crucial for solving many types of algebraic problems. Mastery of linear equations equips you with the skills to tackle more complex mathematical issues.