Solving Linear Equations using the Substitution Method

6 Questions

在使用替换法解线性方程组时，第一步是做什么？

将方程中的一个变量解出来

在使用替换法解$x + 2y = 9$和$3x + 5y = 20$的交点时，第二步是什么？

将第一个方程的解代入第二个方程

在替换法中，最终要做什么才能得到未知数的值？

将一个变量的值代回另一个方程求解

如果在替换法中错误地将两个方程相乘而不是代入，将会导致什么结果？

得到错误的结果

如果在替换法中一开始就将$y = -7$代入$x = 9 - 2y$，会导致什么问题？

$x$ 的值将会是 $-23$

在使用替换法解线性方程组时，为什么有时候比直接通过绘图找到交点更有优势？

替换法适用于复杂方程

Study Notes

Solving Linear Equations using the Substitution Method

The substitution method is a technique used to solve systems of linear equations algebraically. In this approach, you use one equation to find the value of one variable, and then you substitute that value back into the other equation to solve for the remaining variable. This method is particularly useful when dealing with complex equations where finding the intersection points directly through graphing may not be feasible.

Let's consider an example: finding the intersection of two lines represented by the equations x + 2y = 9 and 3x + 5y = 20. Instead of trying to find the intersection point by graphing the lines, we can use the substitution method.

Substitution Method Steps

Solve one of the equations for any variable: In this case, let's solve the first equation for x. Subtracting 2y from both sides, we get x = 9 - 2y.
Substitute the expression found in step 1 into the other equation: Replace all instances of x in the second equation with the expression derived in step 1. So, 3(9 - 2y) + 5y = 20.
Simplify and solve: Expanding and distributing the terms, we get 27 - 6y + 5y = 20. Rearranging, we have -y = 7. Solving for y, we get y = -7.
Substitute the value of y back into the expression found in step 1: Replace all instances of y in the expression x = 9 - 2y with the value of y we found in step 3. So, x = 9 - 2(-7), which simplifies to x = 9 + 14. Therefore, x = 23.

So, the intersection point of the two lines is (23, -7).