Question

Your weekly paycheck is \(12 \%\) more than your coworker's. Your two paychecks total \(\$ 848\). Find the amount of each paycheck.

Short answer

Your coworker's paycheck is approximately $400, and your paycheck is approximately $448.

Step-by-step solution

Set Up the Equations

Let's denote your coworker's paycheck as \( x \) and your paycheck as \( y \). From the problem, we can establish two equations: 1) \( y = 1.12x \) (since your paycheck is 12% more than your coworker's) and 2) \( x + y = 848 \) (since the total of the two paychecks is $848).

Substitution

Substitute the first equation \( y = 1.12x \) into the second equation to solve for \( x \): \( x + 1.12x = 848 \) . This simplifies to \( 2.12x = 848 \).

Solve for x

To solve for \( x \), isolate it by dividing each side of the equation by 2.12: \( x = 848 / 2.12 \) . Using a calculator, we find that \( x \approx 400 \).

Solve for y

Substitute \( x = 400 \) into the first equation to solve for \( y \): \( y = 1.12 * 400 \). This gives us \( y \approx 448 \).

Key concepts

System of Equations

When faced with a percent increase word problem, it's often useful to employ a system of equations to find the solution. A system of equations is a set of two or more equations that have the same set of variables. In our example, we have two unknowns – your paycheck and your coworker's. To find the values of these two unknowns, we need two equations that relate them.

In the given problem, the relationship is defined by two facts: your paycheck is 12% more than your coworker's, and together, they total $848. The system of equations is a powerful tool in algebra that can be solved using various methods, such as substitution, elimination, or graphing. Here, the most efficient method is substitution, which we'll delve into more deeply in the next section. Understanding systems of equations is essential and much more broadly applicable than just to our current context – it's a fundamental concept in algebra that can model and solve a myriad of real-world problems.

Algebraic Substitution

One common method for solving a system of equations is algebraic substitution. This technique is particularly useful in situations where one equation is easy to solve for one variable in terms of another. In our exercise, we define one paycheck using a percentage of the other (12% more), making it a perfect candidate for substitution.

To use this method, solve one of the equations for one variable and then substitute that expression into the other equation. This is what we did in the step-by-step solution when we wrote that your paycheck (\(y\)) is 1.12 times your coworker's paycheck (\(x\)). By substituting this expression for \(y\) into the second equation dealing with the total sum of the paychecks, we're left with an equation in one variable that we can solve. While substitution can involve quite a bit of algebraic manipulation, it simplifies the problem down to an equation that can be solved using basic operations.

Solving Linear Equations

Once we have substituted our variables and have a single equation, as we did with \(2.12x = 848\), we're ready to solve a linear equation. A linear equation is any equation that, when graphed, gives a straight line. They are usually in the form \(ax + b = c\), where \(a\), \(b\), and \(c\) are constants.

Our task is to isolate the variable (in this case, \(x\)) and solve for its value. We do this by performing operations that simplify the equation, maintaining the equality. For the equation \(2.12x = 848\), we divide both sides by 2.12, giving us \(x = 400\). Subsequently, we take this value for \(x\) and substitute it back into our original expression for \(y\) to find the value of the second paycheck. Solving linear equations is an essential skill, not just for algebra but for higher-level math, science, and many applications in technology and finance.