Problem 181 - Elementary Algebra

Question

Solve the system \(\left\\{\begin{array}{l}x+y=10 \\ 5 x+8 y=56\end{array}\right.\) (a) by substitution (b) by graphing \(\odot\) Which method do you prefer? Why?

Step-by-Step Solution

Answer

The solution is (8, 2). Substitution is often preferred for a more exact answer.

Step 1: Solve for one variable in the first equation

Given the first equation: \( x + y = 10 \). Solve for \( x \): \( x = 10 - y \).

Step 2: Substitute into the second equation

Substitute \( x = 10 - y \) into the second equation \( 5x + 8y = 56 \): \( 5(10 - y) + 8y = 56 \).

Step 3: Simplify and solve for y

Simplify the equation: \( 50 - 5y + 8y = 56 \) Combine like terms: \( 3y = 6 \) Solve for \( y \): \( y = 2 \).

Step 4: Solve for x

Substitute \( y = 2 \) back into the equation \( x = 10 - y \): \( x = 10 - 2 \). Solve for \( x \): \( x = 8 \).

Step 5: Verify the solution (optional)

Substitute \( x = 8 \) and \( y = 2 \) into both original equations to verify: First equation: \( 8 + 2 = 10 \), which is true. Second equation: \( 5(8) + 8(2) = 40 + 16 = 56 \), which is also true.

Step 6: Graph the equations

Express each equation in slope-intercept form (\( y = mx + b \)): First equation: \( y = -x + 10 \). Second equation: \( 8y = -5x + 56 \) \( y = -\frac{5}{8}x + 7 \). Plot both lines on a coordinate plane and find their intersection point (8, 2).

Step 7: Discuss preferred method

Preference for method can depend on factors such as ease of use and clarity of finding the solution. Substitution may be preferred if solving algebraically is straightforward, while graphing gives a visual representation but can be less precise.

Key Concepts

Substitution Method

The substitution method is a systematic way to solve systems of linear equations. Here's how it works:
First, solve one of the equations for one variable in terms of the other. For example, in the system given:

Graphing Method

Graphing involves plotting both equations on the same coordinate plane and finding the point where the lines intersect. Each linear equation can be written in the slope-intercept form, which is:

Linear Equations

Linear equations are fundamental in algebra and represent straight lines when graphed on a coordinate plane. They have the general form of: