Problem 87 - College Algebra

Question

Evaluate \(x^{2}-(x y-y)\) for \(x\) satisfying \(\frac{3(x+3)}{5}=2 x+6\) and \(y\) satisfying \(-2 y-10=5 y+18\)

Step-by-Step Solution

Answer

So, the result of evaluating the expression \(x^{2}-(x y-y)\) for the given conditions is 17.

Step 1: Solve for \(x\)

First thing to do to solve \(\frac{3(x+3)}{5}=2 x+6\) is to clear the fraction. This can be done by multiplying the whole equation by 5. That gives: \(3x+9=10x+30\). Then, by rearranging, we get \(10x-3x=30-9\), which simplifies to \(7x=21\). Dividing by 7 gives \(x=3\).

Step 2: Solve for \(y\)

Solve \(-2 y-10=5 y+18\) by first isolating \(y\). This can be done by combining like terms and then dividing, which yields \(-2y-5y=18+10\). Simplify to get \(-7y=28\). And dividing by -7, we arrive at \(y=-4\).

Step 3: Substitute \(x\) and \(y\) values into the expression

With \(x=3\) and \(y=-4\), substitute those values into \(x^{2}-(x y-y)\). We get \(3^2-(3*(-4)-(-4))\), which equals \(9 - (-8) = 17\).

Key Concepts

Evaluate Expressions

Evaluating expressions in algebra involves replacing variables with their corresponding values and then simplifying the expression using the rules of arithmetic and algebra. It's a crucial skill that allows students to find the value of an expression for given values of its variables.

• Begin with the initial expression:
  In our example, we have the expression \(x^{2}-(xy-y)\).
• Replace the variables with given values:
  Here, we substitute \(x=3\) and \(y=-4\) into the expression.
• Simplify using arithmetic rules:
  This means performing the operations as dictated by the order of operations—parentheses, exponents, multiplication and division (from left to right), and addition and subtraction (from left to right).

Following these steps with the given parameters results in the value of 17 for the initial algebraic expression, meaning that once variables are correctly substituted, the process concludes with straightforward arithmetic.

Solving Linear Equations

Solving linear equations is about finding the value of the variable that makes the equation true. These are the steps one usually follows:

• Isolate the variable: Aim to get the variable by itself on one side of the equation.
  In our exercise, to solve \(\frac{3(x+3)}{5}=2x+6\), we multiplied by 5 to clear the fraction and then rearranged the terms to isolate \(x\).
• Combine like terms: Group variables and constants together and simplify. For example, \(3x+9=10x+30\) transitions to \(7x=21\) once like terms are combined.
• Divide both sides of the equation to solve for the variable:
  After combining like terms, we divided both sides by 7 to get \(x=3\).

By following this process, linear equations become manageable, allowing for step-by-step solutions that lead to the clear identification of the value of the variable.

Substitution Method

The substitution method is a technique used to solve systems of equations. This method involves solving one of the equations for one variable and then replacing that variable in the other equation with the obtained value.

• Solve one of the equations for one variable:
  Our example required solving for \(y\) in the equation \(–2y – 10 = 5y + 18\).
• Substitute the value found into the other equation or expression:
  We found \(y = –4\) and then substituted the values of \(x\) and \(y\) into the original algebraic expression.
• Solve the remaining equation or simplify the expression:
  After substituting, we simplified the expression to find the final result.

The strength of the substitution method is that it can turn a complicated system of equations into a simpler one, making it easier to manage and solve.