To determine the set of coordinates that satisfies both equations [tex]\(3x - 2y = 15\)[/tex] and [tex]\(4x - y = 20\)[/tex], we will solve this system of linear equations step-by-step.
1. Set up the equations:
[tex]\[ 3x - 2y = 15 \, \quad (1) \][/tex]
[tex]\[ 4x - y = 20 \quad (2) \][/tex]
2. Solve one equation for one variable:
Let's solve Equation (2) for [tex]\( y \)[/tex].
[tex]\[ 4x - y = 20 \][/tex]
Rearrange to solve for [tex]\( y \)[/tex]:
[tex]\[ y = 4x - 20 \][/tex]
3. Substitute this expression into Equation (1):
Substitute [tex]\( y = 4x - 20 \)[/tex] into [tex]\( 3x - 2y = 15 \)[/tex]:
[tex]\[ 3x - 2(4x - 20) = 15 \][/tex]
4. Simplify and solve for [tex]\( x \)[/tex]:
[tex]\[ 3x - 8x + 40 = 15 \][/tex]
[tex]\[ -5x + 40 = 15 \][/tex]
[tex]\[ -5x = 15 - 40 \][/tex]
[tex]\[ -5x = -25 \][/tex]
[tex]\[ x = 5 \][/tex]
5. Substitute [tex]\( x = 5 \)[/tex] back into the expression for [tex]\( y \)[/tex]:
Using [tex]\( y = 4x - 20 \)[/tex]:
[tex]\[ y = 4(5) - 20 \][/tex]
[tex]\[ y = 20 - 20 \][/tex]
[tex]\[ y = 0 \][/tex]
6. Confirm the solution:
Verify that [tex]\( x = 5 \)[/tex] and [tex]\( y = 0 \)[/tex] satisfy both original equations.
- For [tex]\( 3x - 2y = 15 \)[/tex]:
[tex]\[ 3(5) - 2(0) = 15 \][/tex]
[tex]\[ 15 = 15 \][/tex] (True)
- For [tex]\( 4x - y = 20 \)[/tex]:
[tex]\[ 4(5) - 0 = 20 \][/tex]
[tex]\[ 20 = 20 \][/tex] (True)
Since both equations are satisfied, the set of coordinates [tex]\((5, 0)\)[/tex] is the solution.
Therefore, the correct answer is [tex]\((5, 0)\)[/tex].
