Precalculus Examples

$x = - 7 t + 1$ , $y = - 10 t + 9$

Step 1

Move all terms containing variables to the left.

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Step 1.1

Add $7 t$ to both sides of the equation.

$x + 7 t = 1, y = - 10 t + 9$

Step 1.2

Add $10 t$ to both sides of the equation.

$x + 7 t = 1, y + 10 t = 9$

Step 2

Reorder the polynomial.

$7 t + x = 1$

$y + 10 t = 9$

Step 3

Reorder the polynomial.

$10 t + y = 9$

$7 t + x = 1$

Step 4

Multiply each equation by the value that makes the coefficients of $t$ opposite.

$(- 10) \cdot (7 t + x) = (- 10) (1)$

$(7) \cdot (10 t + y) = (7) (9)$

Step 5

Simplify.

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Step 5.1

Simplify the left side.

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Step 5.1.1

Simplify $(- 10) \cdot (7 t + x)$ .

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Step 5.1.1.1

Apply the distributive property.

$- 10 (7 t) - 10 x = (- 10) (1)$

$(7) \cdot (10 t + y) = (7) (9)$

Step 5.1.1.2

Multiply $7$ by $- 10$ .

$- 70 t - 10 x = (- 10) (1)$

$(7) \cdot (10 t + y) = (7) (9)$

$- 70 t - 10 x = (- 10) (1)$

$(7) \cdot (10 t + y) = (7) (9)$

$- 70 t - 10 x = (- 10) (1)$

$(7) \cdot (10 t + y) = (7) (9)$

Step 5.2

Simplify the right side.

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Step 5.2.1

Multiply $- 10$ by $1$ .

$- 70 t - 10 x = - 10$

$(7) \cdot (10 t + y) = (7) (9)$

$- 70 t - 10 x = - 10$

$(7) \cdot (10 t + y) = (7) (9)$

Step 5.3

Simplify the left side.

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Step 5.3.1

Simplify $(7) \cdot (10 t + y)$ .

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Step 5.3.1.1

Apply the distributive property.

$- 70 t - 10 x = - 10$

$7 (10 t) + 7 y = (7) (9)$

Step 5.3.1.2

Multiply $10$ by $7$ .

$- 70 t - 10 x = - 10$

$70 t + 7 y = (7) (9)$

$- 70 t - 10 x = - 10$

$70 t + 7 y = (7) (9)$

$- 70 t - 10 x = - 10$

$70 t + 7 y = (7) (9)$

Step 5.4

Simplify the right side.

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Step 5.4.1

Multiply $7$ by $9$ .

$- 70 t - 10 x = - 10$

$70 t + 7 y = 63$

$- 70 t - 10 x = - 10$

$70 t + 7 y = 63$

$- 70 t - 10 x = - 10$

$70 t + 7 y = 63$

Step 6

Add the two equations together to eliminate $t$ from the system.

		$-$	$7$	$0$	$t$	$-$	$1$	$0$	$x$	$=$	$-$	$1$	$0$
	$+$		$7$	$0$	$t$					$=$		$6$	$3$
$7$	$y$					$-$	$1$	$0$	$x$	$=$		$5$	$3$

Step 7

Simplify the equation and solve for $x$ .

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Step 7.1

Subtract $7 y$ from both sides of the equation.

$- 10 x = 53 - 7 y$

Step 7.2

Divide each term in $- 10 x = 53 - 7 y$ by $- 10$ and simplify.

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Step 7.2.1

Divide each term in $- 10 x = 53 - 7 y$ by $- 10$ .

$\frac{- 10 x}{- 10} = \frac{53}{- 10} + \frac{- 7 y}{- 10}$

Step 7.2.2

Simplify the left side.

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Step 7.2.2.1

Cancel the common factor of $- 10$ .

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Step 7.2.2.1.1

Cancel the common factor.

$\frac{- 10 x}{- 10} = \frac{53}{- 10} + \frac{- 7 y}{- 10}$

Step 7.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{53}{- 10} + \frac{- 7 y}{- 10}$

Step 7.2.3

Simplify the right side.

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Step 7.2.3.1

Simplify each term.

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Step 7.2.3.1.1

Move the negative in front of the fraction.

$x = - \frac{53}{10} + \frac{- 7 y}{- 10}$

Step 7.2.3.1.2

Dividing two negative values results in a positive value.

$x = - \frac{53}{10} + \frac{7 y}{10}$

Step 8

Substitute the value found for $x$ into one of the original equations, then solve for $t$ .

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Step 8.1

Substitute the value found for $x$ into one of the original equations to solve for $t$ .

$- 70 t - 10 (- \frac{53}{10} + \frac{7 y}{10}) = - 10$

Step 8.2

Simplify each term.

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Step 8.2.1

Apply the distributive property.

$- 70 t - 10 (- \frac{53}{10}) - 10 \frac{7 y}{10} = - 10$

Step 8.2.2

Cancel the common factor of $10$ .

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Step 8.2.2.1

Move the leading negative in $- \frac{53}{10}$ into the numerator.

$- 70 t - 10 (\frac{- 53}{10}) - 10 \frac{7 y}{10} = - 10$

Step 8.2.2.2

Factor $10$ out of $- 10$ .

$- 70 t + 10 (- 1) \frac{- 53}{10} - 10 \frac{7 y}{10} = - 10$

Step 8.2.2.3

Cancel the common factor.

$- 70 t + 10 \cdot - 1 \frac{- 53}{10} - 10 \frac{7 y}{10} = - 10$

Step 8.2.2.4

Rewrite the expression.

$- 70 t - 1 \cdot - 53 - 10 \frac{7 y}{10} = - 10$

Step 8.2.3

Multiply $- 1$ by $- 53$ .

$- 70 t + 53 - 10 \frac{7 y}{10} = - 10$

Step 8.2.4

Cancel the common factor of $10$ .

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Step 8.2.4.1

Factor $10$ out of $- 10$ .

$- 70 t + 53 + 10 (- 1) \frac{7 y}{10} = - 10$

Step 8.2.4.2

Cancel the common factor.

$- 70 t + 53 + 10 \cdot - 1 \frac{7 y}{10} = - 10$

Step 8.2.4.3

Rewrite the expression.

$- 70 t + 53 - 1 (7 y) = - 10$

Step 8.2.5

Multiply $7$ by $- 1$ .

$- 70 t + 53 - 7 y = - 10$

Step 8.3

Move all terms not containing $t$ to the right side of the equation.

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Step 8.3.1

Subtract $53$ from both sides of the equation.

$- 70 t - 7 y = - 10 - 53$

Step 8.3.2

Add $7 y$ to both sides of the equation.

$- 70 t = - 10 - 53 + 7 y$

Step 8.3.3

Subtract $53$ from $- 10$ .

$- 70 t = - 63 + 7 y$

Step 8.4

Divide each term in $- 70 t = - 63 + 7 y$ by $- 70$ and simplify.

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Step 8.4.1

Divide each term in $- 70 t = - 63 + 7 y$ by $- 70$ .

$\frac{- 70 t}{- 70} = \frac{- 63}{- 70} + \frac{7 y}{- 70}$

Step 8.4.2

Simplify the left side.

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Step 8.4.2.1

Cancel the common factor of $- 70$ .

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Step 8.4.2.1.1

Cancel the common factor.

$\frac{- 70 t}{- 70} = \frac{- 63}{- 70} + \frac{7 y}{- 70}$

Step 8.4.2.1.2

Divide $t$ by $1$ .

$t = \frac{- 63}{- 70} + \frac{7 y}{- 70}$

Step 8.4.3

Simplify the right side.

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Step 8.4.3.1

Simplify each term.

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Step 8.4.3.1.1

Cancel the common factor of $- 63$ and $- 70$ .

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Step 8.4.3.1.1.1

Factor $- 7$ out of $- 63$ .

$t = \frac{- 7 (9)}{- 70} + \frac{7 y}{- 70}$

Step 8.4.3.1.1.2

Cancel the common factors.

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Step 8.4.3.1.1.2.1

Factor $- 7$ out of $- 70$ .

$t = \frac{- 7 \cdot 9}{- 7 \cdot 10} + \frac{7 y}{- 70}$

Step 8.4.3.1.1.2.2

Cancel the common factor.

$t = \frac{- 7 \cdot 9}{- 7 \cdot 10} + \frac{7 y}{- 70}$

Step 8.4.3.1.1.2.3

Rewrite the expression.

$t = \frac{9}{10} + \frac{7 y}{- 70}$

Step 8.4.3.1.2

Cancel the common factor of $7$ and $- 70$ .

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Step 8.4.3.1.2.1

Factor $7$ out of $7 y$ .

$t = \frac{9}{10} + \frac{7 (y)}{- 70}$

Step 8.4.3.1.2.2

Cancel the common factors.

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Step 8.4.3.1.2.2.1

Factor $7$ out of $- 70$ .

$t = \frac{9}{10} + \frac{7 y}{7 \cdot - 10}$

Step 8.4.3.1.2.2.2

Cancel the common factor.

$t = \frac{9}{10} + \frac{7 y}{7 \cdot - 10}$

Step 8.4.3.1.2.2.3

Rewrite the expression.

$t = \frac{9}{10} + \frac{y}{- 10}$

Step 8.4.3.1.3

Move the negative in front of the fraction.

$t = \frac{9}{10} - \frac{y}{10}$

Step 9

This is the final solution to the independent system of equations.

$x = - \frac{53}{10} + \frac{7 y}{10}$

$t = \frac{9}{10} - \frac{y}{10}$

		$-$	$7$	$0$	$t$	$-$	$1$	$0$	$x$	$=$	$-$	$1$	$0$
	$+$		$7$	$0$	$t$					$=$		$6$	$3$
$7$	$y$					$-$	$1$	$0$	$x$	$=$		$5$	$3$

		$-$	$7$	$0$	$t$	$-$	$1$	$0$	$x$	$=$	$-$	$1$	$0$
	$+$		$7$	$0$	$t$					$=$		$6$	$3$
$7$	$y$					$-$	$1$	$0$	$x$	$=$		$5$	$3$

		$-$	$7$	$0$	$t$	$-$	$1$	$0$	$x$	$=$	$-$	$1$	$0$
	$+$		$7$	$0$	$t$					$=$		$6$	$3$
$7$	$y$					$-$	$1$	$0$	$x$	$=$		$5$	$3$