Problem 64 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use any method to solve the system. \(\left\\{\begin{aligned} 7 x+3 y &=16 \\ y &=x+2 \end{aligned}\right.\)

Step-by-Step Solution

Answer

The solution to the system of equations is \(x = 1\) and \(y = 3\).

Step 1: Substitute \(y = x + 2\) into the first equation

We can place \(x + 2\) for \(y\) in the first equation to simplify the equation further. The equation \(7x + 3y = 16\) therefore becomes \(7x + 3(x + 2) = 16\).

Step 2: Simplify the resulting equation to solve for \(x\)

After substitution, the equation becomes \(7x + 3x + 6 = 16\). By combining like terms, this simplifies to \(10x = 10\). Dividing by 10 on both sides yields \(x = 1\).

Step 3: Substitute \(x = 1\) into the second equation to solve for \(y\)

Substitute \(x = 1\) into the equation \(y = x + 2\), resulting in \(y = 1 + 2\). This yields \(y = 3\).

Key Concepts

Simultaneous Equations

Simultaneous equations are two or more equations with multiple variables that are true at the same time. In our exercise, we have the system of equations below:

\[ \begin{align*} 7x + 3y &= 16, \ y &= x + 2 \end{align*} \]

This means both conditions must be satisfied by the same variables \(x\) and \(y\). Simultaneous equations often arise in real-world situations, such as determining the break-even point in business or finding the intersection of two graphs.

To solve simultaneous equations, we need to find the specific values of the variables that make both equations true. Techniques such as substitution and elimination are commonly used to solve these equations efficiently. Understanding how simultaneous equations work allows us to solve complex mathematical problems easily.

Substitution Method

The substitution method is a technique used to solve simultaneous equations. It involves solving one equation for a single variable and substituting that expression into the other equation.

In our given system, since the second equation is already solved for \(y\), which is\( y = x + 2 \), this makes substitution straightforward.

• Step 1: Substitute \(y = x + 2\) into the first equation \(7x + 3y = 16\).
  This gives us \(7x + 3(x + 2) = 16\), replacing \(y\) with \(x + 2\).

By substituting, we reduce the number of variables and simplify the problem. This method is particularly useful when one of the equations in the system is already solved for a variable, as it reduces the complexity of solving the system.

Solving for Variables

After substitution, we streamline the process of solving for a single variable at a time. Let's recap the solving process:

• After substituting, the equation simplifies to \(7x + 3x + 6 = 16\).
• Here, we combine like terms to get \(10x + 6 = 16\).
• Subtract 6 from both sides: \(10x = 10\).
• Divide both sides by 10 to find \(x\): \(x = 1\).

This value of \(x\) makes the first equation true.

Now, we use \(x = 1\) to solve for \(y\) incorporating the second equation.

• Substituting \(x\) into \(y = x + 2\), we get \(y = 1 + 2\).
• This results in \(y = 3\).

Thus, the solution \(x = 1, y = 3\) satisfies both equations, meaning both initial equations are simultaneously true using these values. This orderly procedure highlights systematic ways of finding solutions to linear systems.