Question Video: Solving Simultaneous Equations by Substitution | Nagwa Question Video: Solving Simultaneous Equations by Substitution | Nagwa

Question Video: Solving Simultaneous Equations by Substitution Mathematics • Third Year of Preparatory School

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Solve the following system of equations if possible: 𝑦 = π‘₯ + 1, 𝑦 = π‘₯ βˆ’ 9. [A] π‘₯ = βˆ’1, 𝑦 = βˆ’9 [B] π‘₯ = βˆ’9, 𝑦 = 1 [C] π‘₯ = 9, 𝑦 = 1 [D] π‘₯ = 1, 𝑦 = 9 [E] There are no solutions.

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Video Transcript

Solve the following system of equations if possible: 𝑦 equals π‘₯ plus one and 𝑦 equals π‘₯ minus nine. The possible solutions are (a) π‘₯ equals negative one and 𝑦 equals negative nine. (b) π‘₯ equals negative nine and 𝑦 equals one. (c) π‘₯ equals nine and 𝑦 equals one. (d) π‘₯ equals one and 𝑦 equals nine. Or (e) there are no solutions.

To solve a system of equations like the one we have here means we need to find the values of the two variables, in this case π‘₯ and 𝑦, that satisfy both equations simultaneously. The question states that we should solve this system of equations if possible because it could be the case that there are in fact no values of π‘₯ and 𝑦 that satisfy both equations, in which case there will be no solutions. Looking at the two equations, we can see that in each case, we’ve been given an expression for one variable, 𝑦, in terms of the other variable, π‘₯. If there is to be a solution to this pair of equations, then the value of 𝑦 must be the same in both. So we can equate the two expressions that we’ve been given for 𝑦 in terms of π‘₯. Doing so gives the equation π‘₯ plus one is equal to π‘₯ minus nine.

To attempt to solve this equation, we want to collect all of the π‘₯-terms on the same side. So we can subtract π‘₯ from each side as this will eliminate the π‘₯ on the right-hand side. But doing so not only eliminates the π‘₯ on the right-hand side of the equation, but it also eliminates the π‘₯ on the left-hand side, leaving one is equal to negative nine. Now, this of course is a contradiction. One is certainly not equal to negative nine. And this tells us that it isn’t possible for the 𝑦-values in the two equations to be the same for the same π‘₯-value. This means then that there are no solutions to this system of equations.

We can demonstrate why this is the case using our knowledge of straight-line graphs. We know that if the equation of a straight line is given in the slope–intercept form, 𝑦 equals π‘šπ‘₯ plus 𝑏, then the value of π‘š gives the slope of the line and the value of 𝑏 gives its 𝑦-intercept. The line represented by the first equation 𝑦 equals π‘₯ plus one, therefore, has a slope of one and also a 𝑦-intercept of one. Its graph will look a little something like this. The straight line represented by the second equation also has a slope of one, but it has a 𝑦-intercept of negative nine. So its graph will look a little something like this.

We can see from our sketch, but also from our working, that the two lines have the same slope. They are, in fact, parallel to one another, and parallel lines never intersect. If the straight lines represented by these two equations never intersect, then there is no solution to their simultaneous equations. Our answer then is (e) there are no solutions.

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