Problem 18 - A Graphical Approach to College Algebra

Question

Solve each system by substitution. $$\begin{aligned}&8 x+3 y=2\\\&5 x=17+6 y\end{aligned}$$

Step-by-Step Solution

Answer

The solution is \(x = 1\) and \(y = -2\).

Step 1: Solve One Equation for One Variable

Let's start by solving the second equation for the variable \(x\). The equation is given as:\[ 5x = 17 + 6y \]We can solve for \(x\) by isolating it on one side:\[ x = \frac{17 + 6y}{5} \]

Step 2: Substitution of Expression for x

Now that we have \(x\) in terms of \(y\), substitute this expression into the first equation:\[ 8x + 3y = 2 \]Substitute \(x = \frac{17 + 6y}{5}\) into the equation:\[ 8\left(\frac{17 + 6y}{5}\right) + 3y = 2 \]

Step 3: Simplify and Solve for y

Multiply through by 5 to eliminate the fraction:\[ 8(17 + 6y) + 3y \times 5 = 2 \times 5 \]This simplifies to:\[ 136 + 48y + 15y = 10 \]Combine like terms:\[ 63y + 136 = 10 \]Subtract 136 from both sides:\[ 63y = -126 \]Divide by 63:\[ y = -2 \]

Step 4: Substitute y to Find x

Now use the value of \(y\) in the expression for \(x\):\[ x = \frac{17 + 6(-2)}{5} \]Simplify inside the parenthesis:\[ x = \frac{17 - 12}{5} \]\[ x = \frac{5}{5} \]\[ x = 1 \]

Step 5: Conclusion

The solution to the system of equations is:\[ x = 1 \] and \[ y = -2 \]. This means the two lines intersect at point \((1, -2)\).

Key Concepts

System of Equations

A system of equations consists of two or more equations with the same set of unknowns. In our example, the system includes two equations: \[8x + 3y = 2\]and \[5x = 17 + 6y\]. These equations represent two straight lines on a graph. The solution to the system is the point where these two lines intersect.

• If the equations have a common solution, the lines intersect at a single point.
• If the lines are parallel and do not meet, there is no solution.
• If the lines coincide, there are infinitely many solutions.

Understanding systems of equations involves recognizing how different equations relate to each other and finding their common solutions, if they exist.
In this case, we have found that the intersection point or the solution for the system is the point \((1, -2)\).

Linear Equations

Linear equations are algebraic equations of the first degree. This means that the variables in a linear equation are not raised to any power other than one. Each term in a linear equation is either a constant or the product of a constant and a single variable. For instance, in the equation\(8x + 3y = 2\),both terms,\(8x\) and \(3y\), follow this rule. Linear equations form straight lines when graphed on a coordinate plane. The coefficients of the variables affect the slope and position of the line. In particular, linear equations are fundamental in mathematics and are used extensively to model real-world scenarios.
This line's behavior can easily be analyzed, and solutions can often be found by simple algebraic operations.

Algebraic Solution

An algebraic solution involves using algebraic methods to find the unknowns in an equation or a system of equations. For the problem at hand, the substitution method is employed as the algebraic approach. The substitution method involves the following steps:

• Solve one of the equations for one variable in terms of the others. In this case: \(x = \frac{17 + 6y}{5}\).
• Substitute this expression into the other equation, replacing the variable.
• Simplify to find the value of one variable. Here we find \(y = -2\) by simplifying.
• Substitute back to find the other variable. Once \(y\) is known, we solve for \(x\) leading to \(x = 1\).

This method provides exact solutions by systematically isolating variables and substituting expressions. In doing so, we can determine the exact point where the given equations intersect.