Question

solve the system by elimination. {4x - 5y = 25, -3x - 2y = 10} one solution: blank no solution infinite number of solutions question help: message instructor

Explanation:

Step1: Eliminate a variable

To eliminate \( y \), we'll make the coefficients of \( y \) opposites. Multiply the first equation \( 4x - 5y = 25 \) by \( 2 \) and the second equation \( -3x - 2y = 10 \) by \( 5 \).
First equation after multiplying: \( 2(4x - 5y)=2(25)\) gives \( 8x - 10y = 50 \).
Second equation after multiplying: \( 5(-3x - 2y)=5(10)\) gives \( -15x - 10y = 50 \).

Step2: Subtract the equations

Subtract the second new equation from the first:
\( (8x - 10y) - (-15x - 10y)=50 - 50 \)
Simplify: \( 8x - 10y + 15x + 10y = 0 \)
Combine like terms: \( 23x = 0 \)? Wait, no, wait: Wait, \( 8x - (-15x)=23x \), \( -10y - (-10y)=0 \), \( 50 - 50 = 0 \). Wait, that can't be right. Wait, no, I made a mistake. Wait, the first equation multiplied by 2: \( 8x - 10y = 50 \), the second multiplied by 5: \( -15x - 10y = 50 \). Now, subtract the second from the first: \( (8x - 10y) - (-15x - 10y) = 50 - 50 \) → \( 8x -10y +15x +10y = 0 \) → \( 23x = 0 \)? Wait, no, 50 - 50 is 0, so \( 23x = 0 \)? Wait, that would mean \( x = 0 \), but let's check again. Wait, maybe I should add instead of subtract? Wait, no, the coefficients of \( y \) are both -10y, so to eliminate \( y \), we can subtract the two equations. Wait, first equation: \( 8x -10y = 50 \), second: \( -15x -10y = 50 \). Subtract second from first: \( (8x -10y) - (-15x -10y) = 50 - 50 \) → \( 8x +15x -10y +10y = 0 \) → \( 23x = 0 \) → \( x = 0 \). Then plug \( x = 0 \) into the first original equation: \( 4(0) -5y =25 \) → \( -5y =25 \) → \( y = -5 \). Wait, but let's check with the second equation: \( -3(0) -2(-5)=10 \) → \( 0 +10 =10 \), which works. Wait, so my initial subtraction was correct. Wait, \( 23x = 0 \) → \( x=0 \), then \( y = -5 \). Wait, but let's redo the elimination correctly. Alternatively, multiply the first equation by 3 and the second by 4 to eliminate \( x \). First equation: \( 4x -5y =25 \) ×3: \( 12x -15y =75 \). Second equation: \( -3x -2y =10 \) ×4: \( -12x -8y =40 \). Now add the two equations: \( (12x -15y) + (-12x -8y)=75 +40 \) → \( -23y =115 \) → \( y = -5 \). Then plug \( y = -5 \) into first equation: \( 4x -5(-5)=25 \) → \( 4x +25 =25 \) → \( 4x =0 \) → \( x=0 \). There we go, that's better. So the solution is \( x=0 \), \( y=-5 \).

Answer:

\((0, -5)\)