Problem 13 - Algebra 1: Concepts and Skills

Question

Tell which equation you would use to isolate a variable. Explain. $$ \begin{aligned} 3 x-2 y &=19 \\ x+y &=8 \end{aligned} $$

Step-by-Step Solution

Answer

To isolate a variable, the second equation 'x + y = 8' is used. 'x' is expressed in terms of 'y' as 'x = 8 - y', which is then substituted into the first equation to solve for 'y'. The value of 'y' is then substituted back into the equation 'x = 8 - y' to solve for 'x'.

Step 1: Identify Variable to Isolate

From the system of equations, it is easier to isolate 'x' in the second equation, as it already has a coefficient of 1. Use the second equation \(x + y = 8\).

Step 2: Isolate the Variable

Express 'x' in terms of 'y' from the second equation. This can be done by subtracting 'y' from both sides of the equation, leading to the result \(x = 8 - y\).

Step 3: Substitute and Solve

Now, substitute this value for 'x' into the first equation \(3x - 2y = 19\), which becomes: \(3(8 - y) - 2y = 19\). Simplify this equation and solve for 'y'.

Step 4: Find the Value of the Other Variable

Once the value of 'y' is attained, substitute it back into the second equation \(x = 8 - y\), to find the value of 'x'. Solve the equation to get the value of 'x'.

Key Concepts

Isolating Variables

Isolating a variable is a fundamental technique in algebra that allows us to find the value of an unknown in an equation. This process involves manipulating the equation so that the variable we want to solve for is on one side by itself. In our exercise with the equations \(3x - 2y = 19\) and \(x + y = 8\), isolating a variable is best done with the second equation, \(x + y = 8\), because 'x' has a coefficient of 1, making it simpler to isolate.

To isolate 'x', you subtract 'y' from both sides of the equation, which results in \(x = 8 - y\). This now 'isolated' variable 'x' can then be used in the substitution method to find the values of 'x' and 'y'. Remember, the goal is to perform operations that will simplify the equation without changing the equality, such as adding, subtracting, multiplying, or dividing both sides of the equation by the same number.

Substitution Method

The substitution method is a powerful tool in algebra used to solve systems of equations. It involves first isolating a variable in one equation and then substituting that expression into the other equation. Here's a more in-depth look using our exercise's equations.

Once we have isolated 'x' to \(x = 8 - y\), we substitute this expression into the first equation in place of 'x'. Therefore, \(3(8-y) - 2y = 19\) is what we obtain. Our next step is to distribute and combine like terms to solve for 'y'. After determining the value of 'y', we can substitute it back into the isolated equation to find the value of 'x'. This back and forth process is the essence of the substitution method, permitting us to solve for multiple variables efficiently.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and arithmetic operations like addition and subtraction. In the context of our system of equations, \(3x - 2y = 19\) and \(x + y = 8\), each side of the equation forms an algebraic expression.

In solving systems of equations, understanding how to manipulate these expressions is crucial. For instance, when we substituted \(x = 8 - y\) into the first equation, we engaged in an algebraic operation that adjusted the expression. Knowing how to simplify and combine like terms—such as adding together terms with 'y' or distributing multipliers—are pivotal skills in solving equations. This command of algebraic expressions enables us to transition from complex systems to simple equations we can solve.