Math Problem Statement

Find m and c in 2/3 x + 3/4y = 1 1/2

Solution

We are given the equation:

23x+34y=32\frac{2}{3}x + \frac{3}{4}y = \frac{3}{2}

Our task is to express this in the slope-intercept form, y=mx+cy = mx + c, where mm is the slope and cc is the y-intercept.

Step 1: Isolate yy on one side

We start by isolating the yy-term on the left-hand side. First, subtract 23x\frac{2}{3}x from both sides:

34y=23x+32\frac{3}{4}y = -\frac{2}{3}x + \frac{3}{2}

Step 2: Solve for yy

Next, multiply both sides of the equation by 43\frac{4}{3} to eliminate the fraction in front of yy:

y=43(23x+32)y = \frac{4}{3}\left(-\frac{2}{3}x + \frac{3}{2}\right)

Now, distribute the 43\frac{4}{3} across the terms inside the parentheses:

y=(4323)x+(4332)y = \left(\frac{4}{3} \cdot -\frac{2}{3}\right)x + \left(\frac{4}{3} \cdot \frac{3}{2}\right)

Simplify each term:

y=89x+2y = -\frac{8}{9}x + 2

Step 3: Identify mm and cc

Now that the equation is in the form y=mx+cy = mx + c:

  • The slope m=89m = -\frac{8}{9}
  • The y-intercept c=2c = 2

Final Answer:

  • m=89m = -\frac{8}{9}
  • c=2c = 2

Would you like more details or have any questions?

Here are some related questions to explore:

  1. How do you find the slope given two points?
  2. What is the general method for converting a linear equation to slope-intercept form?
  3. How can you determine the x-intercept from this equation?
  4. What does the slope of a line represent geometrically?
  5. How can we verify the solution by graphing?

Tip: When converting to slope-intercept form, always aim to isolate yy first for clarity!

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Math Problem Analysis

Mathematical Concepts

Algebra
Linear Equations
Slope-Intercept Form

Formulas

Slope-Intercept Form: y = mx + c

Theorems

Conversion to Slope-Intercept Form

Suitable Grade Level

Grades 8-10