Solve {l}{1.4-3.2t=5/9.8k}{(0.6+0.8t)*9.8=2.5k}

$$\left\{ \begin{array} { l } { 1.4 - 3.2 t = \frac { 5 } { 9.8 } k } \\ { ( 0.6 + 0.8 t ) \times 9.8 = 2.5 k } \end{array} \right.$$

Solve for t, k

$$t=\frac{1}{24}\approx 0.041666667$$

$$k = \frac{931}{375} = 2\frac{181}{375} \approx 2.482666667$$

Steps Using Substitution

Consider the first equation. Expand \frac{5}{9.8} by multiplying both numerator and the denominator by 10.

$$1.4-3.2t=\frac{50}{98}k$$

Reduce the fraction \frac{50}{98} to lowest terms by extracting and canceling out 2.

$$1.4-3.2t=\frac{25}{49}k$$

Subtract \frac{25}{49}k from both sides.

$$1.4-3.2t-\frac{25}{49}k=0$$

Subtract 1.4 from both sides. Anything subtracted from zero gives its negation.

$$-3.2t-\frac{25}{49}k=-1.4$$

Consider the second equation. Subtract 2.5k from both sides.

$$\left(0.6+0.8t\right)\times 9.8-2.5k=0$$

To solve a pair of equations using substitution, first solve one of the equations for one of the variables. Then substitute the result for that variable in the other equation.

$$-3.2t-\frac{25}{49}k=-1.4,9.8\left(0.8t+0.6\right)-2.5k=0$$

Choose one of the equations and solve it for t by isolating t on the left hand side of the equal sign.

$$-3.2t-\frac{25}{49}k=-1.4$$

Add \frac{25k}{49} to both sides of the equation.

$$-3.2t=\frac{25}{49}k-1.4$$

Divide both sides of the equation by -3.2, which is the same as multiplying both sides by the reciprocal of the fraction.

$$t=-0.3125\left(\frac{25}{49}k-1.4\right)$$

Multiply -0.3125 times \frac{25k}{49}-\frac{7}{5}.

$$t=-\frac{125}{784}k+0.4375$$

Substitute -\frac{125k}{784}+\frac{7}{16} for t in the other equation, 9.8\left(0.8t+0.6\right)-2.5k=0.

$$9.8\left(0.8\left(-\frac{125}{784}k+0.4375\right)+0.6\right)-2.5k=0$$

Multiply 0.8 times -\frac{125k}{784}+\frac{7}{16}.

$$9.8\left(-\frac{25}{196}k+0.35+0.6\right)-2.5k=0$$

Add 0.35 to 0.6 by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$9.8\left(-\frac{25}{196}k+0.95\right)-2.5k=0$$

Multiply 9.8 times -\frac{25k}{196}+\frac{19}{20}.

$$-\frac{5}{4}k+9.31-2.5k=0$$

Add -\frac{5k}{4} to -\frac{5k}{2}.

$$-\frac{15}{4}k+9.31=0$$

Subtract 9.31 from both sides of the equation.

$$-\frac{15}{4}k=-9.31$$

Divide both sides of the equation by -\frac{15}{4}, which is the same as multiplying both sides by the reciprocal of the fraction.

$$k=\frac{931}{375}$$

Substitute \frac{931}{375} for k in t=-\frac{125}{784}k+0.4375. Because the resulting equation contains only one variable, you can solve for t directly.

$$t=-\frac{125}{784}\times \frac{931}{375}+0.4375$$

Multiply -\frac{125}{784} times \frac{931}{375} by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$t=-\frac{19}{48}+0.4375$$

Add 0.4375 to -\frac{19}{48} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$t=\frac{1}{24}$$

The system is now solved.

$$t=\frac{1}{24},k=\frac{931}{375}$$

Steps Using Matrices

Consider the first equation. Expand \frac{5}{9.8} by multiplying both numerator and the denominator by 10.

$$1.4-3.2t=\frac{50}{98}k$$

Reduce the fraction \frac{50}{98} to lowest terms by extracting and canceling out 2.

$$1.4-3.2t=\frac{25}{49}k$$

Subtract \frac{25}{49}k from both sides.

$$1.4-3.2t-\frac{25}{49}k=0$$

Subtract 1.4 from both sides. Anything subtracted from zero gives its negation.

$$-3.2t-\frac{25}{49}k=-1.4$$

Consider the second equation. Subtract 2.5k from both sides.

$$\left(0.6+0.8t\right)\times 9.8-2.5k=0$$

Put the equations in standard form and then use matrices to solve the system of equations.

$$-3.2t-\frac{25}{49}k=-1.4,9.8\left(0.8t+0.6\right)-2.5k=0$$

Simplify the second equation to put it in standard form.

$$9.8\left(0.8t+0.6\right)-2.5k=0$$

Multiply 9.8 times 0.8t+0.6.

$$7.84t+5.88-2.5k=0$$

Subtract 5.88 from both sides of the equation.

$$7.84t-2.5k=-5.88$$

Write the equations in matrix form.

$$\left(\begin{matrix}-3.2&-\frac{25}{49}\\7.84&-2.5\end{matrix}\right)\left(\begin{matrix}t\\k\end{matrix}\right)=\left(\begin{matrix}-1.4\\-5.88\end{matrix}\right)$$

Left multiply the equation by the inverse matrix of \left(\begin{matrix}-3.2&-\frac{25}{49}\\7.84&-2.5\end{matrix}\right).

$$inverse(\left(\begin{matrix}-3.2&-\frac{25}{49}\\7.84&-2.5\end{matrix}\right))\left(\begin{matrix}-3.2&-\frac{25}{49}\\7.84&-2.5\end{matrix}\right)\left(\begin{matrix}t\\k\end{matrix}\right)=inverse(\left(\begin{matrix}-3.2&-\frac{25}{49}\\7.84&-2.5\end{matrix}\right))\left(\begin{matrix}-1.4\\-5.88\end{matrix}\right)$$

The product of a matrix and its inverse is the identity matrix.

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}t\\k\end{matrix}\right)=inverse(\left(\begin{matrix}-3.2&-\frac{25}{49}\\7.84&-2.5\end{matrix}\right))\left(\begin{matrix}-1.4\\-5.88\end{matrix}\right)$$

Multiply the matrices on the left hand side of the equal sign.

$$\left(\begin{matrix}t\\k\end{matrix}\right)=inverse(\left(\begin{matrix}-3.2&-\frac{25}{49}\\7.84&-2.5\end{matrix}\right))\left(\begin{matrix}-1.4\\-5.88\end{matrix}\right)$$

For the 2\times 2 matrix \left(\begin{matrix}a&b\\c&d\end{matrix}\right), the inverse matrix is \left(\begin{matrix}\frac{d}{ad-bc}&\frac{-b}{ad-bc}\\\frac{-c}{ad-bc}&\frac{a}{ad-bc}\end{matrix}\right), so the matrix equation can be rewritten as a matrix multiplication problem.

$$\left(\begin{matrix}t\\k\end{matrix}\right)=\left(\begin{matrix}-\frac{2.5}{-3.2\left(-2.5\right)-\left(-\frac{25}{49}\times 7.84\right)}&-\frac{-\frac{25}{49}}{-3.2\left(-2.5\right)-\left(-\frac{25}{49}\times 7.84\right)}\\-\frac{7.84}{-3.2\left(-2.5\right)-\left(-\frac{25}{49}\times 7.84\right)}&-\frac{3.2}{-3.2\left(-2.5\right)-\left(-\frac{25}{49}\times 7.84\right)}\end{matrix}\right)\left(\begin{matrix}-1.4\\-5.88\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}t\\k\end{matrix}\right)=\left(\begin{matrix}-\frac{5}{24}&\frac{25}{588}\\-\frac{49}{75}&-\frac{4}{15}\end{matrix}\right)\left(\begin{matrix}-1.4\\-5.88\end{matrix}\right)$$

Multiply the matrices.

$$\left(\begin{matrix}t\\k\end{matrix}\right)=\left(\begin{matrix}-\frac{5}{24}\left(-1.4\right)+\frac{25}{588}\left(-5.88\right)\\-\frac{49}{75}\left(-1.4\right)-\frac{4}{15}\left(-5.88\right)\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}t\\k\end{matrix}\right)=\left(\begin{matrix}\frac{1}{24}\\\frac{931}{375}\end{matrix}\right)$$

Extract the matrix elements t and k.

$$t=\frac{1}{24},k=\frac{931}{375}$$