Problem 14 - Algebra 2

Question

Solve each system of equations by using substitution. \(5 a-b=17\) \(3 a+2 b=5\)

Step-by-Step Solution

Answer

The solution is \( a = 3 \) and \( b = -2 \).

Step 1: Substitute

First, we'll solve the first equation for one of the variables. Let's solve for \( b \): \( b = 5a - 17 \). Substitute this expression into the second equation \( 3a + 2b = 5 \) to use the substitution method.

Step 2: Substitute Expression in Second Equation

Substitute \( b = 5a - 17 \) into the second equation: \( 3a + 2(5a - 17) = 5 \).

Step 3: Simplify the Equation

Distribute \( 2 \) in the equation: \( 3a + 10a - 34 = 5 \). Combine like terms to simplify: \( 13a - 34 = 5 \).

Step 4: Isolate the Variable

Add \( 34 \) to both sides of the equation: \( 13a = 39 \).

Step 5: Solve for \( a \)

Divide both sides by \( 13 \): \( a = 3 \).

Step 6: Substitute Back to Find \( b \)

Now substitute \( a = 3 \) back into the expression for \( b \): \( b = 5(3) - 17 \).

Step 7: Simplify to Find \( b \)

Calculate \( b = 15 - 17 \): \( b = -2 \).

Key Concepts

Substitution Method

The substitution method is an approach to solving a system of equations. This method is particularly useful when one equation in the system can easily be solved for one variable.
Here's how it works:

• First, solve one of the equations for one of the variables. This means expressing that variable in terms of the other.
• Then, substitute this expression into the other equation. This substitution helps in reducing the system from two equations with two variables to one equation with a single variable, making it much easier to solve.

In our example, we solved for the variable \( b \) in terms of \( a \), i.e., \( b = 5a - 17 \), and substituted this expression into the second equation \( 3a + 2b = 5 \). This is the essence of the substitution method, helping simplify the problem significantly.

Solving Linear Equations

Solving linear equations involves finding the values of the variables that make the equation true. A linear equation is, essentially, an equation whose graph forms a straight line. It typically appears in the form \( ax + by = c \).

• Each term in the equation is either a constant or a product of a constant and a single variable.
• The degree of each variable is always one, which means the variable should not have an exponent other than one.

In the substitution method, once we've made a substitution, we end up with a linear equation in one variable. For instance, after substituting \( b = 5a - 17 \) into the second equation, we have the linear equation \( 3a + 10a - 34 = 5 \). Solving this equation involves combining like terms and following steps to isolate and solve for the variable.

Algebraic Expressions

An algebraic expression consists of constants, variables, and operations like addition, subtraction, multiplication, and division. Understanding how to manipulate algebraic expressions is crucial in solving systems of equations.

• In our example, \( 5a - 17 \) is an algebraic expression showing the relationship between \( a \) and \( b \) after isolating one variable.
• Carefully substituting algebraic expressions can help simplify complex equations into an easier format, suitable for further solving.

Remember, when performing operations with algebraic expressions, the forces welcome: simplifying through distributive property and combining like terms. This skill is pivotal in ensuring your equations remain manageable, especially in systems of equations.

Variable Isolation

Variable isolation is a fundamental step in solving an equation. It involves getting the variable of interest alone on one side of the equation, which is critical in solving for its value. Typically, variable isolation includes these steps:

• Use operations like addition, subtraction, multiplication, or division to both sides of the equation to eliminate other terms from the side with the variable.
• Be systematic to ensure every step maintains the balance of the equation.

For example, in isolating \( a \) from \( 13a - 34 = 5 \), adding \( 34 \) to both sides gives \( 13a = 39 \). Finally, divide by \( 13 \) to solve for \( a \). Isolating variables systematically ensures accuracy and makes solving systems of equations possible.