Which Answer Can Be Used to Find the Solution to the System of Equations?

Solving Systems of Equations: Which Method to Utilize?

Systems of equations are multiple equations that all take a common solution. Students run into these systems of equations when at that place are multiple 'unknowns' - or variables - that have non been given to them yet. When this happens, the goal for students is to utilise given information in the equations to solve for all variables.

To solve systems of equations, it is helpful for students to take a background agreement of simple algebraic equations, variables, and graphing linear equations.

How do I solve a system of equations?

In that location are three methods used to solve systems of equations: graphing, exchange, and elimination.

To solve a organization by graphing, you lot but graph the given equations and notice the point(south) where they all intersect. The coordinate of this point will give you the values of the variables that you are solving for. This is most efficient when the equations are already written in slope-intercept course.

The next method is exchange. Exchange is best used when one of the equations is in terms of one of the variables such every bit y=2x+4 , but the equations tin always be manipulated. The starting time step in this method is to solve one of the equations for one variable. Once an expression for the variable is establish, substitute or plug in the expression into the other equation where the original variable was to solve for the number value of the next variable. The final step is to substitute the number value that was constitute in for its corresponding variable in the original equation.

The tertiary method is elimination. Elimination is adding the equations together in gild to create an equation with only i variable. This tin merely be done when the coefficients of i variable in both equations are opposites and will cancel each other out one time added together. Emptying is best used when this is already occurring in the equations, only the equations tin can too exist manipulated into creating mutual coefficients by either multiplying or dividing equations by a certain number. The side by side footstep would be to apply the equation that we created to find the value of the variable and then plug that value back into an original equation to observe the remaining variable.

Here's an instance of a problem where solving a system of equations is necessary:

Logan has answered 0.8 as many math questions as Castilian questions, and he'due south answered 5 more than English questions than Spanish questions. If Logan has answered 33 questions in total, how many math questions has Logan answered?

How do I solve this trouble?

The first pace is to create equations from the discussion problem. To do this, we must assign variables to each unknown role of the problem. The variables x, y, and z will represent the amount of math, Spanish, and English questions that Logan has answered respectively.

Equally Logan has answered 0.8 times as many math questions as Spanish ones, the equation to correspond this would be 0.8y=x . The second equation would be z=y+v to represent how Logan has answered five more English questions than Spanish ones. The final equation would be x+y+z=33 to correspond how Logan answered 33 questions in total.

Looking dorsum at the original question, the goal of this problem is to find how many math questions Logan has answered. Since the first equation nosotros found was 0.8y=10 , nosotros can see that we only need the y variable to find the value of x, or the amount of math questions that were answered. Because there are two equations already solved in terms of two variables, 0.8y=x and z=y+5 , exchange would be the nigh efficient method. To do this method, we would substitute these equations for the x and y variables of the third equation which would give united states of america (0.8y)+y+(y+5)=33 .

The next step would be to solve this equation for the y variable by combining like terms: 2.8y=28 which would give us y=10 or 10 Spanish questions answered. Now that we have found the value of the y variable, we can plug information technology back into the equation, 0.8y=10 , to detect the value of x. Past substituting y with its value of x, we would get 0.8(10)=10 which would requite the states a value of 8 for x.

What's the answer?

Logan answered 8 math questions.

What concepts did nosotros use?

To solve that example problem, nosotros used several different math concepts. The commencement one we used was how to write equations from give-and-take problems. Through our understanding of the problem, we were able to assign each unknown aspect of the trouble a variable and then create equations based off of their relationships in the trouble which nosotros would so recognize equally a organisation of equations.

The 2d concept that nosotros used is solving the system. The equations weren't written in slope-intercept grade, and then graphing would not take been an efficient method. There were no variables that were opposite each other, and then we too ruled out the elimination method. By recognizing substitution every bit the best method to apply, nosotros were able to efficiently use our math skills to solve for the unknown variables in a system of equations.

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