Problem 87 - College Algebra

Question

Explain how to solve a system of equations using the substitution method. Use \(y-3=3 x\) and \(3 x+4 y=6\) to illustrate your explanation.

Step-by-Step Solution

Answer

The solution to the system of equations is \(-0.4, 2.8\)

Step 1: Solve one equation for one variable

The first equation \(y-3=3x\) can be solved for \(y\), it becomes \(y=3x+3\)

Step 2: Substitute into the other equation

We substitute \(y=3x+3\) into \(3x+4y=6\). It becomes \(3x + 4(3x+3) = 6\)

Step 3: Solve for x

To solve the equation for \(x\), the equation is simplified to \(3x+12x+12=6\), so \(15x= -6\), and finally \(x = -6/15 = -0.4\)

Step 4: Solve for y

We can substitute \(x=-0.4\) into the first equation so \(y = 3*-0.4+3 = 2.8\)

Key Concepts

Substitution Method

The substitution method is a powerful algebraic technique used to solve systems of equations. In this method, one equation is solved for one variable, and this expression is substituted into the other equation. The aim is to simplify the system to a single variable equation that can be easily solved.

Here's how it works:

• Identify one equation and isolate one variable. This makes it easier to express one variable in terms of the other.
• Substitute this expression into the other equation. You replace the isolated variable in the second equation with the expression from the first equation.
• Solve for the remaining variable. This newly formed equation will now contain only one variable which can be directly solved.
• Use the solved variable to find the value of the other variable using the expression from the first equation.

Through this method, you convert a system of equations into something more manageable, making it much simpler to find the solution.

Algebraic Equations

Algebraic equations are mathematical statements that express equality between two algebraic expressions. These equations can include numbers, variables, and arithmetic operations that represent relationships between quantities.

There are different types of algebraic equations. In the context of solving systems of equations, we often deal with linear equations. Understanding each part of an algebraic equation helps in breaking down more complex problems into manageable parts. It involves recognizing terms, coefficients, and operations within the equation.

In our system:

• The first equation is transformed to show how one quantity depends on the other. It's expressed as: \(y = 3x + 3\).
• The second equation remains \(3x + 4y = 6\), which reveals relationships between \(x\) and \(y\).

Recognizing these forms helps in utilizing techniques like substitution method to find solutions.

Linear Equations

Linear equations are a specific type of algebraic equation where the highest power of any variable is one. These equations graph as straight lines when plotted on a coordinate plane and often model simple relationships between variables.

Key characteristics include:

• They have constant rates of change, meaning the slope is a constant value.
• They form a straight line on a graph, which is why they’re called 'linear.'
• The standard form of a linear equation in two variables is \( ax + by = c \), where \(a\), \(b\), and \(c\) are constants.

In our example, both equations \(y - 3 = 3x\) and \(3x + 4y = 6\) are linear. They represent planes when plotted and intersect at a single point which is the solution to the system of equations.

Solution Steps

The solution steps for solving systems of equations using the substitution method are like a well-organized recipe. Following each step carefully ensures the correct solution is reached:

Step 1: Solve for one Variable
In the first equation \(y - 3 = 3x\), isolate \(y\) by adding 3 to both sides to get \(y = 3x + 3\).

Step 2: Substitute
Replace \(y\) in the second equation \(3x + 4y = 6\) with \(3x + 3\). This translates the equation to \(3x + 4(3x + 3) = 6\).

Step 3: Solve for x
Simplify and solve the equation \(3x + 12x + 12 = 6\), leading to \(15x = -6\). Solve for \(x\) to get \(x = -0.4\).

Step 4: Solve for y
Use the value of \(x\) in \(y = 3x + 3\). Substitute \(-0.4\) for \(x\), giving \(y = 2.8\).

These steps guide solving the system, where each step builds upon the previous one leading to the solution \((x, y) = (-0.4, 2.8)\).