Question

antonio graphs these equations and finds that the lines intersect at a single point, (-5, 0.25).
description \tmath
equation a: 4y - 3x = 16
equation b: -x - 8y = 3
which statement is true about the values x = -5 and y = 0.25?
a. they satisfy equation b but not equation a.
b. they are the only values that make both equations true.
c. they satisfy equation a but not equation b.
d. they show that the equations represent the same line.

Explanation:

Step1: Check Equation A

Substitute \( x = -5 \) and \( y = 0.25 \) into Equation A: \( 4y - 3x = 16 \).
Left - hand side (LHS) \(= 4\times0.25-3\times(-5)\)
\(= 1 + 15\)
\(= 16\), which is equal to the right - hand side (RHS) of Equation A. So \( x=-5,y = 0.25 \) satisfies Equation A.

Step2: Check Equation B

Substitute \( x = -5 \) and \( y = 0.25 \) into Equation B: \( -x-8y = 3 \).
Left - hand side (LHS) \(=-(-5)-8\times0.25\)
\(= 5 - 2\)
\(= 3\), which is equal to the right - hand side (RHS) of Equation B. So \( x = -5,y=0.25 \) satisfies Equation B.

Step3: Analyze the nature of the solution

Since the two lines intersect at a single point \((-5,0.25)\), this point is the only solution that satisfies both equations simultaneously.

• Option A: Since the point satisfies both equations, this option is wrong.
• Option B: The intersection point of two non - parallel lines (if they are not the same line) is the only solution that makes both equations true. Since the lines intersect at a single point, this point \((-5,0.25)\) is the only solution for the system of equations, so this option is correct.
• Option C: Since the point satisfies both equations, this option is wrong.
• Option D: If the equations represented the same line, they would have infinitely many solutions. But here they intersect at a single point, so they are not the same line. This option is wrong.

Answer:

B. They are the only values that make both equations true