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3. Systems of Linear Equations & Inequalities
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United States of AmericaFL
Grade 912

3.02 Solving systems of equations by substitution

Lesson
Practice
Lesson

Concept summary

When at least one equation in a system of equations can be solved quickly for one variable, the system can be solved efficiently by using substitution.

Substitution method

A method of solving a system of equations by replacing a variable in one equation with an equivalent expression from another equation

If after solving a system of equations the result is always true, independent of the variables, then the system has infinitely many solutions. If the result is always false, independent of the variables, then the system has no solutions.

Worked examples

Example 1

Solve the following system of equations using the substitution method: \begin{cases}y=x+11\\y=3x+19 \end{cases}

Approach

We first want to number our equations to make it easier to work with.

1\displaystyle y\displaystyle =\displaystyle x + 11
2\displaystyle y\displaystyle =\displaystyle 3x + 19

Since both equations already have y isolated, we can start by substituting equation 1 into equation 2 to eliminate y from the equation. We can then solve for x, and substitute this value back into one of the equations to solve for y.

Solution

\displaystyle x + 11\displaystyle =\displaystyle 3x + 19Substitute equation 1 into equation 2
\displaystyle 11\displaystyle =\displaystyle 2x + 19Subtract x from both sides
\displaystyle -8\displaystyle =\displaystyle 2xSubtract 19 from both sides
\displaystyle -4\displaystyle =\displaystyle xDivide both sides by 2
\displaystyle y\displaystyle =\displaystyle -4 + 11Substitute back into equation 1
\displaystyle y\displaystyle =\displaystyle 7Evaluate the addition

So the solution to the system of equations is x = -4, y = 7.

Example 2

A mother is currently 7 times as old as her son. In 3 years time, she will be 5 times as old as him.

a

Write a system of equations for this scenario, where y represents the mother's current age and x represents the current age of her son.

Solution

1\displaystyle y\displaystyle =\displaystyle 7x
2\displaystyle y + 3\displaystyle =\displaystyle 5\left(x + 3\right)
b

Solve the system of equations for their ages.

Approach

Since 1 is already in the form y = ⬚, we can substitute 1 into 2 without rearrangement.

Solution

\displaystyle 7x + 3\displaystyle =\displaystyle 5\left(x + 3\right)Substitute equation 1 into equation 2
\displaystyle 7x + 3\displaystyle =\displaystyle 5x + 15Distribute the multiplication
\displaystyle 2x + 3\displaystyle =\displaystyle 15Subtract 5x from both sides
\displaystyle 2x\displaystyle =\displaystyle 12Subtract 3 from both sides
\displaystyle x\displaystyle =\displaystyle 6Divide both sides by 2
\displaystyle y\displaystyle =\displaystyle 7\left(6\right)Substitute back into equation 1
\displaystyle y\displaystyle =\displaystyle 42Evaluate the multiplication

So the mother's age is 42 and her son's age is 6.

c

Does the solution make sense in terms of the context? Explain your answer.

Solution

Yes. 6 years earlier, when her son was born, the mother would have been 36 years old. It makes sense that the mother is older than the son, and both values are valid ages.

Outcomes

MA.912.AR.1.2

Rearrange equations or formulas to isolate a quantity of interest.

MA.912.AR.9.1

Given a mathematical or real-world context, write and solve a system of two-variable linear equations algebraically or graphically.

MA.912.AR.9.6

Given a real-world context, represent constraints as systems of linear equations or inequalities. Interpret solutions to problems as viable or non-viable options.

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