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Sagot :
Answer:
[tex]\displaystyle\mathsf{Solution:\:x=-\frac{13}{11},\:y=\frac{36}{11}\quad\ or \quad \Bigg(-\frac{13}{11},\:\:\frac{36}{11}\:\:\Bigg)}[/tex]
Step-by-step explanation:
We are given the following systems of linear equations with two variables:
[tex]\displaystyle\mathsf{\left \{{{Equation\:1:\quad y\:=\:4x+8} \atop {Equation\:2:\quad y=-7x-5}} \right. }[/tex]
We will use the substitution method to solve for the solution. In order to determine the solution to the given system, it is important to solve one of the equations for one of the variables.
Solution:
Step 1: Substitute the value of y from Equation 2 (-7x - 5) into Equation 1 :
Equaton 2: y = -7x - 5
Equation 1: y = 4x + 8
⇒ -7x - 5 = 4x + 8
Step 2: Combine like terms:
Subtract 8 from both sides ⇒ -7x - 5 - 8 = 4x + 8 - 8
⇒ -7x - 13 = 4x
Add 7x to both sides ⇒ -7x + 7x - 13 = 4x + 7x
⇒ - 13 = 11x
Step 3: Divide both sides by 11 to isolate x.
⇒ [tex]\displaystyle\mathsf{\frac{-13}{11}\:=\:\frac{11x}{11}}[/tex]
⇒ [tex]\displaystyle\bf{x\:=\:-\frac{13}{11}}[/tex]
Step 4: Substitute [tex]\displaystyle\mathsf{{x\:=\:-\frac{13}{11}}}[/tex] into Equation 2 to solve for y:
⇒ [tex]\displaystyle\mathsf{{Equation\:2:\quad y\:=-7x\:-\:5}}[/tex]
[tex]\displaystyle\mathsf{{\Rightarrow\ \:Equation\:2:\quad y\:=\:-7\Bigg[-\frac{13}{11}}\Bigg]\:-\:5}[/tex]
[tex]\displaystyle\mathsf{{\Rightarrow\ \:Equation\:2:\quad y\:=\:\Bigg[\frac{-7\:\: \times\ -13}{11}}\Bigg]\:-\:5}[/tex]
[tex]\displaystyle\mathsf{{\Rightarrow\ \:Equation\:2:\quad y\:=\:\Bigg[\frac{91}{11}}\Bigg]\:-\:5}[/tex]
Step 5: Covert -5 into fraction
[tex]\displaystyle\mathsf{{\Rightarrow\ y\:=\:\Bigg[\frac{91}{11}}\Bigg]\:-\:\Bigg[\frac{11\times\ 5}{11\div\ 1}\Bigg]}[/tex]
[tex]\displaystyle\mathsf{{\Rightarrow\ y\:=\:\frac{91}{11}}-\:\frac{55}{11}}[/tex]
[tex]\displaystyle\mathsf{{\Rightarrow\ y\:=\:\frac{36}{11}}}[/tex][tex]\fbox{Solution:\:(-\frac{13}{11},\frac{36}{11}}[/tex]
[tex]\displaystyle\mathsf{Solution:\:\Bigg(-\frac{13}{11},\:\:\frac{36}{11}\:\:\Bigg)}[/tex]
Step 6: Check this solution in both equations:
[tex]\displaystyle\mathsf{\left \{{{Equation\:1:\quad y\:=\:4x+8} \atop {Equation\:2:\quad y=-7x-5}} \right. }[/tex]
Equation 1: y = 4x + 8
[tex]\displaystyle\mathsf{\Rightarrow\ Equation\:1:\quad \frac{36}{11} =\:4\Bigg[ -\frac{13}{11} \Bigg] +8}[/tex]
[tex]\displaystyle\mathsf{\Rightarrow\ Equation\:1:\quad \frac{36}{11} =\:\Bigg[ \frac{4\:\: \times\ -13 }{11} \Bigg] +8}[/tex]
[tex]\displaystyle\mathsf{\Rightarrow\ Equation\:1:\quad \frac{36}{11} =\:\Bigg[ \frac{-52}{11} \Bigg] + \Bigg[ \frac{11\times\ 8 }{11 \div\ 1} \Bigg]}[/tex]
[tex]\displaystyle\mathsf{\Rightarrow\ Equation\:1:\quad \frac{36}{11} =\:\Bigg[ \frac{-52}{11} \Bigg] + \Bigg[ \frac{88 }{11} \Bigg]}[/tex]
[tex]\displaystyle\mathsf{\Rightarrow\ Equation\:1:\quad \frac{36}{11} =\:\frac{36}{11}}[/tex]
Equaton 2: y = -7x - 5
[tex]\displaystyle\mathsf{\Rightarrow\ Equation\:2:\quad \frac{36}{11} =\:-7\Bigg[ -\frac{13}{11} \Bigg] -5}[/tex]
[tex]\displaystyle\mathsf{{\Rightarrow\ \:Equation\:2:\quad \frac{36}{11}\:=\:\Bigg[\frac{-7\:\: \times\ -13}{11}}\Bigg]\:-\:5}[/tex]
[tex]\displaystyle\mathsf{{\Rightarrow\ Equation\:2:\quad \frac{36}{11}\:=\:\Bigg[\frac{91}{11}}\Bigg]\:-\:\Bigg[\frac{11\times\ 5}{11\div\ 1}\Bigg]}[/tex]
[tex]\displaystyle\mathsf{{\Rightarrow\ Equation\:2:\quad \frac{36}{11}\:=\:\frac{91}{11}}-\:\frac{55}{11}}[/tex]
[tex]\displaystyle\mathsf{\Rightarrow\ Equation\:2:\quad \frac{36}{11} =\:\frac{36}{11}}[/tex]
Conclusion:
After substituting our derived values for x and y in Step 6, we can confirm that x = -13/11 and y = 36/11 satisfy both equations in the given system.
Therefore, the following is solution set of the given system: [tex]\displaystyle\mathsf{Solution:\:x=-\frac{13}{11},\:y=\frac{36}{11}\quad\ or \quad \Bigg(-\frac{13}{11},\:\:\frac{36}{11}\:\:\Bigg)}[/tex]
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Keywords:
Linear equations
Linear functions
Systems of linear equations
Slope-intercept form
Substitution method
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