Answer :
To determine which of the given equations are equivalent to [tex]\( y = \frac{2}{3} x - 4 \)[/tex] when written in slope-intercept form (i.e., [tex]\(y = mx + b\)[/tex]), we'll convert each given equation to slope-intercept form by isolating [tex]\( y \)[/tex] and comparing it to the given equation [tex]\( y = \frac{2}{3} x - 4 \)[/tex].
1. Equation: [tex]\( 3x - 2y = 4 \)[/tex]
- Isolate [tex]\( y \)[/tex]:
[tex]\[ 3x - 2y = 4 \implies -2y = -3x + 4 \implies y = \frac{3}{2}x - 2 \][/tex]
2. Equation: [tex]\( 2x - 3y = 12 \)[/tex]
- Isolate [tex]\( y \)[/tex]:
[tex]\[ 2x - 3y = 12 \implies -3y = -2x + 12 \implies y = \frac{2}{3}x - 4 \][/tex]
3. Equation: [tex]\( -4(2x - 3y) = -4(12) \)[/tex]
- Simplify and isolate [tex]\( y \)[/tex]:
[tex]\[ -8x + 12y = -48 \implies 12y = 8x - 48 \implies y = \frac{2}{3}x - 4 \][/tex]
4. Equation: [tex]\( 2(x + 6) = 3y \)[/tex]
- Expand and isolate [tex]\( y \)[/tex]:
[tex]\[ 2(x + 6) = 3y \implies 2x + 12 = 3y \implies y = \frac{2}{3}x + 4 \][/tex]
5. Equation: [tex]\( 2x - 3y = 4 \)[/tex]
- Isolate [tex]\( y \)[/tex]:
[tex]\[ 2x - 3y = 4 \implies -3y = -2x + 4 \implies y = \frac{2}{3}x - \frac{4}{3} \][/tex]
Now, we compare the slope-intercept forms of each equation to the given equation [tex]\( y = \frac{2}{3} x - 4 \)[/tex]:
- Equation 1: [tex]\( y = \frac{3}{2} x - 2 \)[/tex] is not equivalent to [tex]\( y = \frac{2}{3} x - 4 \)[/tex].
- Equation 2: [tex]\( y = \frac{2}{3} x - 4 \)[/tex] is equivalent to [tex]\( y = \frac{2}{3} x - 4 \)[/tex].
- Equation 3: [tex]\( y = \frac{2}{3} x - 4 \)[/tex] is equivalent to [tex]\( y = \frac{2}{3} x - 4 \)[/tex].
- Equation 4: [tex]\( y = \frac{2}{3} x + 4 \)[/tex] is not equivalent to [tex]\( y = \frac{2}{3} x - 4 \)[/tex].
- Equation 5: [tex]\( y = \frac{2}{3} x - \frac{4}{3} \)[/tex] is not equivalent to [tex]\( y = \frac{2}{3} x - 4 \)[/tex].
So, the equations that are equivalent to [tex]\( y = \frac{2}{3} x - 4 \)[/tex] are:
[tex]\[ \boxed{2 \text{ and } 3} \][/tex]
1. Equation: [tex]\( 3x - 2y = 4 \)[/tex]
- Isolate [tex]\( y \)[/tex]:
[tex]\[ 3x - 2y = 4 \implies -2y = -3x + 4 \implies y = \frac{3}{2}x - 2 \][/tex]
2. Equation: [tex]\( 2x - 3y = 12 \)[/tex]
- Isolate [tex]\( y \)[/tex]:
[tex]\[ 2x - 3y = 12 \implies -3y = -2x + 12 \implies y = \frac{2}{3}x - 4 \][/tex]
3. Equation: [tex]\( -4(2x - 3y) = -4(12) \)[/tex]
- Simplify and isolate [tex]\( y \)[/tex]:
[tex]\[ -8x + 12y = -48 \implies 12y = 8x - 48 \implies y = \frac{2}{3}x - 4 \][/tex]
4. Equation: [tex]\( 2(x + 6) = 3y \)[/tex]
- Expand and isolate [tex]\( y \)[/tex]:
[tex]\[ 2(x + 6) = 3y \implies 2x + 12 = 3y \implies y = \frac{2}{3}x + 4 \][/tex]
5. Equation: [tex]\( 2x - 3y = 4 \)[/tex]
- Isolate [tex]\( y \)[/tex]:
[tex]\[ 2x - 3y = 4 \implies -3y = -2x + 4 \implies y = \frac{2}{3}x - \frac{4}{3} \][/tex]
Now, we compare the slope-intercept forms of each equation to the given equation [tex]\( y = \frac{2}{3} x - 4 \)[/tex]:
- Equation 1: [tex]\( y = \frac{3}{2} x - 2 \)[/tex] is not equivalent to [tex]\( y = \frac{2}{3} x - 4 \)[/tex].
- Equation 2: [tex]\( y = \frac{2}{3} x - 4 \)[/tex] is equivalent to [tex]\( y = \frac{2}{3} x - 4 \)[/tex].
- Equation 3: [tex]\( y = \frac{2}{3} x - 4 \)[/tex] is equivalent to [tex]\( y = \frac{2}{3} x - 4 \)[/tex].
- Equation 4: [tex]\( y = \frac{2}{3} x + 4 \)[/tex] is not equivalent to [tex]\( y = \frac{2}{3} x - 4 \)[/tex].
- Equation 5: [tex]\( y = \frac{2}{3} x - \frac{4}{3} \)[/tex] is not equivalent to [tex]\( y = \frac{2}{3} x - 4 \)[/tex].
So, the equations that are equivalent to [tex]\( y = \frac{2}{3} x - 4 \)[/tex] are:
[tex]\[ \boxed{2 \text{ and } 3} \][/tex]
