Math Problem Statement

Solve the system of equations using the substitution method. y = -1 + (3/8)x 2x - 5y = 6

Solution

Let's solve the given system of equations using the substitution method. The equations are:

  1. y=1+38xy = -1 + \frac{3}{8}x
  2. 2x5y=62x - 5y = 6

Step 1: Substitute yy into the second equation

From the first equation, yy is expressed in terms of xx:
y=1+38xy = -1 + \frac{3}{8}x Substitute this into the second equation, 2x5y=62x - 5y = 6:
2x5(1+38x)=62x - 5(-1 + \frac{3}{8}x) = 6

Step 2: Simplify the equation

Expand 5(1+38x)-5(-1 + \frac{3}{8}x):
2x+5158x=62x + 5 - \frac{15}{8}x = 6 Combine like terms:
(2158)x+5=6\left(2 - \frac{15}{8}\right)x + 5 = 6 Convert 22 to a fraction with a denominator of 8:
168158=18\frac{16}{8} - \frac{15}{8} = \frac{1}{8} So the equation becomes:
18x+5=6\frac{1}{8}x + 5 = 6

Step 3: Solve for xx

Subtract 5 from both sides:
18x=1\frac{1}{8}x = 1 Multiply through by 8:
x=8x = 8

Step 4: Solve for yy

Substitute x=8x = 8 into the first equation, y=1+38xy = -1 + \frac{3}{8}x:
y=1+38(8)y = -1 + \frac{3}{8}(8) Simplify:
y=1+3=2y = -1 + 3 = 2

Final Answer:

The solution is: (x,y)=(8,2)(x, y) = (8, 2)

Would you like further details on any step? Here are related questions for practice:

  1. How would you solve this system of equations using the elimination method?
  2. What happens if one equation cannot be expressed explicitly for substitution?
  3. Can systems of equations always have a single solution? Why or why not?
  4. What does it mean if the equations are dependent or inconsistent?
  5. How can you graphically interpret the solution to this system?

Tip: Always simplify fractions and verify by substituting the solution back into the original equations.

Ask a new question for Free

By Image

Drop file here or Click Here to upload

Math Problem Analysis

Mathematical Concepts

Algebra
Systems of Linear Equations
Substitution Method

Formulas

y = mx + b for substitution
Simplification of linear equations

Theorems

Properties of Equality
Linear Equation Principles

Suitable Grade Level

Grades 8-10