7B: Solving Systems with Substitution

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What does a "solution to a system of equations" mean?

• It's the point where both equations equal zero

• It's the point where graphs of both equations cross the y-axis

• It's the point that is true for both equations at the same time

• It's the point where graphs of both equations cross the x-axis

Answer explanation

A solution to a system of equations is a point that will work in all equations within the system.

How many solutions does the graph have?

• One solution

• No solution

• Infinite solutions

Answer explanation

Parallel lines have the same slope. They will never intersect, so they have no common solution.

How many solutions does the graph have?

• One solution

• No solution

• Infinite solutions

Answer explanation

These lines intersect at ONE point. This means that only one point will be true for the equations of both lines.

How many solutions does the graph have?

• One solution

• No solution

• Infinite solutions

Answer explanation

The lines overlap everywhere, so they have the same equation.

This means the solutions for one equation are also the solutions for the other.

The lines go on infinitely in both directions, so there are an infinite number of points that are shared solutions.

Which answer has a correct first step for solving this system without graphing:

3x + 3y = 3

y = 3x + 5

• The variable is already isolated, so we've already solved the system.

• The variable is already isolated for y, so we substitute it in the first equation and solve for x

  3x + 3(3x + 5) = 3

• Free sha vaca doo

• The variable is already isolated for y, so we substitute it in the second equation and solve for x

  y = 3(3x + 3y) = 3

Answer explanation

$3x+3y=3$

$y=3x+5$
A variable is already isolated here, so we already have an equivalent expression from the second equation that can be substituted in place of y in the first equation.

Solve the following systems of equations using substitution:

x = 6

y = 2x - 3

• (6, 6)

• (6, 9)

• (9, 6)

• (9, 9)

Answer explanation

Solve the following systems of equations using substitution:

5x - 2y = 3

y = 2x

• (6, 3)

• (1, 2)

• (3, 6)

• (2, 1)

Answer explanation

When solving a system of equations algebraically, which statement results in no solution?

• 4 = 7

• -2 = 2

• -3 = 3

• all of the above

Answer explanation

You will be left with a contradiction (something false) when trying to solve a system of equations, if there is no solution.

When solving a system of equations algebraically, which statement results in infinite solutions?

• 4 = 7

• 2 = 2

• -3 = 3

• all of the above

Answer explanation

You will be left with an identity (something true) when trying to solve a system of equations, if there are infinitely many solutions.