cross-multiply

Examples of cross-multiply in the following topics:

• Rational Equations

  • For an equation that involves two fractions or rational expressions, cross-multiplying is a helpful strategy for simplifying the equation or determining the value of a variable.
  • Isolate the variable on the left by multiplying both sides by $8$:

• Solving Equations with Rational Expressions; Problems Involving Proportions

  • by factoring the denominators,we find that we must multiply the left side of the equation by $\displaystyle \frac {x(x-2)}{x(x-2)}$ and the right side of the equation by $\displaystyle \frac {x+6}{x+6}$ , giving
  • Based on the rule above—since the denominators are equal, we can now assume the numerators are equal, so we know that $3x(x-2)= 4x(x+6)$ or, multiplied out, that $3x^2-6x=4x^2+24x$
  • Solve equations with rational expressions (proportions) by finding the LCD or by cross-multipliation

• Mendel's Law of Independent Assortment

  • We then multiply the values along each forked path to obtain the F2 offspring probabilities .
  • The values along each forked pathway can be multiplied because each gene assorts independently.
  • Therefore, multiplying this fraction for each of the four genes, (1/4) × (1/4) × (1/4) × (1/4), we determine that 1/256 of the offspring will be quadruply homozygous recessive.
  • The forked-line method can be used to analyze a trihybrid cross.
  • The probability for each possible combination of traits is calculated by multiplying the probability for each individual trait.

• Volume

  • Volume is measured geometrically by multiplying an object's three dimensions—usually taken as length, width and height.
  • The volume of a cylinder: the cross-sectional area times the height of the cylinder.
  • Measuring cups, as seen in , work by taking a known cross sectional area of a cup and multiplying that by a variable height.
  • Since liquid will always cover the cross section (if there is enough liquid), adding more liquid will increase the height inside the container.

• Lagrange Multiplers

  • This is the method of Lagrange multipliers.
  • Where the Lagrange multiplier $\lambda=0$ we can have a local extremum and the two contours cross instead of meeting tangentially.
  • Therefore where the constraint $g=c$ crosses the contour line $f=-1$, is a local minimum of $f$ on the constraint.
  • The trace and the contour $f=-1$ cross at the minimum as we can see in the figure.
  • Since both $g_x \neq 0$ and $g_y \neq 0$, the Lagrange multiplier $\lambda = 0$ at the minimum.

• Zeroes of Linear Functions

  • Graphically, where the line crosses the $x$-axis, is called a zero, or root.  
  • If there is a horizontal line through any point on the $y$-axis, other than at zero, there are no zeros, since the line will never cross the $x$-axis.  
  • Because the $x$-intercept (zero) is a point at which the function crosses the $x$-axis, it will have the value $(x,0)$, where $x$ is the zero.
  • To find the zero of a linear function, simply find the point where the line crosses the $x$-axis.
  • Subtract $2$ and then multiply by $2$, to obtain:

• Rules for Exponent Arithmetic

  • If you multiply this quantity by $a^n$, i.e. by $n$ additional factors of $a$, then you have $a^{m+n}$ factors in total.
  • If you have $n$ factors of $a$ in the denominator, then you can cross out $n$ factors from the numerator.
  • You can multiply numbers in any order you please.
  • Instead of multiplying together $n$ factors equal to $ab$, you could multiply all of the $a$s together and all of the $b$s together and then finish by multiplying $a^n$ by $b^n$.
  • These two terms can be combined by applying the rule for multiplying exponential expressions with the same base:

• Rules of Probability for Mendelian Inheritance

  • The rules of probability can be applied to Mendelian crosses to determine the expected phenotypes and genotypes of offspring.
  • Using large numbers of crosses, Mendel was able to calculate probabilities and use these to predict the outcomes of other crosses.
  • It states that the probability of two independent events occurring together can be calculated by multiplying the individual probabilities of each event occurring alone.
  • In a genetic cross, the probability of the dominant trait being expressed is dependent upon its frequency.
  • Calculate the probability of traits of pea plants using Mendelian crosses

• Cofactors, Minors, and Further Determinants

  • Cross out the entries that lie in the corresponding row $i$ and column $j$.
  • We will find the determinant of the following matrix A by calculating the determinants of its cofactors for the third, rightmost column and then multiplying them by the elements of that column.
  • The same process is carried out to find the determinants of $C_{13}$ and $C_{33}$, which are then multiplied by $a_{13}$ and $a_{33}$, respectively.

• The Multiplier Effect

  • When the fiscal multiplier exceeds one, the resulting impact on the national income is called the multiplier effect.
  • In economics, the fiscal multiplier is the ratio of change in the national income in relation to the change in government spending that causes it (not to be confused with the monetary multiplier).
  • When the fiscal multiplier exceeds one, the resulting impact on the national income is called the multiplier effect.
  • Although the multiplier effect usually measures values of one, there have been cases where multipliers of less than one are measured.
  • During recessions, the government can use the multiplier effect in order to stimulate the economy.