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1. 已知 a+3ba+b=b+3cb+c=c+3ac+a=m\frac{a+3 b}{a+b}=\frac{b+3 c}{b+c}=\frac{c+3 a}{c+a}=m, 一次函数 y=kx+my=k x+m ...
Mar 9, 2024
1. 已知 a+3ba+b=b+3cb+c=c+3ac+a=m\frac{a+3 b}{a+b}=\frac{b+3 c}{b+c}=\frac{c+3 a}{c+a}=m, 一次函数 y=kx+my=k x+m 的图象过点 (1,3)(1,3), 则一次函数 y=kx+my=k x+m 的解析式是 (a+b+c≠0)\quad(a+b+c \neq 0).
Solution by Steps
step 1
Given the system of equations: a+3ba+b=b+3cb+c=c+3ac+a=m \frac{a+3b}{a+b} = \frac{b+3c}{b+c} = \frac{c+3a}{c+a} = m , we need to find the value of m m
step 2
From the asksia-ll calculation list, we have the set of equations: {ac=b2,bc=a2,b(m2−3)=a(1−m2),ab=c2,c(m2−3)=b(1−m2),c(m2−1)=a(3−m2)} \{a c = b^2, b c = a^2, b (m^2 - 3) = a (1 - m^2), a b = c^2, c (m^2 - 3) = b (1 - m^2), c (m^2 - 1) = a (3 - m^2)\}
step 3
Simplifying the equations, we get m2=2aa+c+1 m^2 = \frac{2a}{a+c} + 1
step 4
The linear equation y=kx+m y = kx + m passes through the point (1,3) (1,3) , which gives us 3=k(1)+m 3 = k(1) + m
step 5
From the asksia-ll calculation list for the second equation, we have k=y−mx k = \frac{y - m}{x} and y(1)=3 y(1) = 3
step 6
Substituting y(1)=3 y(1) = 3 into the equation for k k , we get k=3−m1=3−m k = \frac{3 - m}{1} = 3 - m
step 7
Therefore, the equation of the line is y=(3−m)x+m y = (3 - m)x + m
Answer
The equation of the line is y=(3−m)x+m y = (3 - m)x + m
Key Concept
System of Equations and Linear Functions
Explanation
The key concept is solving a system of equations to find the value of m m and then using the point-slope form of a linear equation to find the slope k k given a point through which the line passes.
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