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GB-20-02-001 · Scaled elimination

Solve the simultaneous equations
2x+5y=−412x+5y=-41
5x+8y=−805x+8y=-80

Simultaneous Equations: Linear (Scaling Coefficients) Booklet p. 27
Method
1) Label both equations
2) Make the first-letter coefficients equal
3) Subtract the entire second equation, watching touching signs
4) Solve for the second letter
5) Substitute back to find the first
Worked Example (GB-20-02-001)
1)
Label the equations: 2x+5y=−41(1)2x + 5y=-41\quad(1) and 5x+8y=−80(2)5x + 8y=-80\quad(2).
2)
Match the first-letter coefficients: (1) × 5, (2) × 2.
3)
10x+25y=−20510x+16y=−160(25−(16))y=−205−(−160)\begin{array}{rcl}10x + 25y&=&-205\\ 10x + 16y&=&-160\\\hline (25-(16))y&=&-205-(-160)\end{array}
4)
Subtract the entire second equation, including its signs: 9y=−45⇒y=−59y=-45\Rightarrow y=-5.
5)
Substitute into (1): 2x+(5)(−5)=−41⇒2x=−16⇒x=−82x+(5)(-5)=-41\Rightarrow 2x=-16\Rightarrow x=-8.
Ans = x=−8,  y=−5x=-8,\;y=-5